• 4.7 Applied Optimization Problems
  • Introduction
  • 1.1 Review of Functions
  • 1.2 Basic Classes of Functions
  • 1.3 Trigonometric Functions
  • 1.4 Inverse Functions
  • 1.5 Exponential and Logarithmic Functions
  • Key Equations
  • Key Concepts
  • Review Exercises
  • 2.1 A Preview of Calculus
  • 2.2 The Limit of a Function
  • 2.3 The Limit Laws
  • 2.4 Continuity
  • 2.5 The Precise Definition of a Limit
  • 3.1 Defining the Derivative
  • 3.2 The Derivative as a Function
  • 3.3 Differentiation Rules
  • 3.4 Derivatives as Rates of Change
  • 3.5 Derivatives of Trigonometric Functions
  • 3.6 The Chain Rule
  • 3.7 Derivatives of Inverse Functions
  • 3.8 Implicit Differentiation
  • 3.9 Derivatives of Exponential and Logarithmic Functions
  • 4.1 Related Rates
  • 4.2 Linear Approximations and Differentials
  • 4.3 Maxima and Minima
  • 4.4 The Mean Value Theorem
  • 4.5 Derivatives and the Shape of a Graph
  • 4.6 Limits at Infinity and Asymptotes
  • 4.8 L’Hôpital’s Rule
  • 4.9 Newton’s Method
  • 4.10 Antiderivatives
  • 5.1 Approximating Areas
  • 5.2 The Definite Integral
  • 5.3 The Fundamental Theorem of Calculus
  • 5.4 Integration Formulas and the Net Change Theorem
  • 5.5 Substitution
  • 5.6 Integrals Involving Exponential and Logarithmic Functions
  • 5.7 Integrals Resulting in Inverse Trigonometric Functions
  • 6.1 Areas between Curves
  • 6.2 Determining Volumes by Slicing
  • 6.3 Volumes of Revolution: Cylindrical Shells
  • 6.4 Arc Length of a Curve and Surface Area
  • 6.5 Physical Applications
  • 6.6 Moments and Centers of Mass
  • 6.7 Integrals, Exponential Functions, and Logarithms
  • 6.8 Exponential Growth and Decay
  • 6.9 Calculus of the Hyperbolic Functions
  • A | Table of Integrals
  • B | Table of Derivatives
  • C | Review of Pre-Calculus

Learning Objectives

  • 4.7.1 Set up and solve optimization problems in several applied fields.

One common application of calculus is calculating the minimum or maximum value of a function. For example, companies often want to minimize production costs or maximize revenue. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume. In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter.

Solving Optimization Problems over a Closed, Bounded Interval

The basic idea of the optimization problems that follow is the same. We have a particular quantity that we are interested in maximizing or minimizing. However, we also have some auxiliary condition that needs to be satisfied. For example, in Example 4.32 , we are interested in maximizing the area of a rectangular garden. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. However, what if we have some restriction on how much fencing we can use for the perimeter? In this case, we cannot make the garden as large as we like. Let’s look at how we can maximize the area of a rectangle subject to some constraint on the perimeter.

Example 4.32

Maximizing the area of a garden.

A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing for the other three sides ( Figure 4.62 ). Given 100 100 ft of wire fencing, determine the dimensions that would create a garden of maximum area. What is the maximum area?

Let x x denote the length of the side of the garden perpendicular to the rock wall and y y denote the length of the side parallel to the rock wall. Then the area of the garden is

We want to find the maximum possible area subject to the constraint that the total fencing is 100 ft . 100 ft . From Figure 4.62 , the total amount of fencing used will be 2 x + y . 2 x + y . Therefore, the constraint equation is

Solving this equation for y , y , we have y = 100 − 2 x . y = 100 − 2 x . Thus, we can write the area as

Before trying to maximize the area function A ( x ) = 100 x − 2 x 2 , A ( x ) = 100 x − 2 x 2 , we need to determine the domain under consideration. To construct a rectangular garden, we certainly need the lengths of both sides to be positive. Therefore, we need x > 0 x > 0 and y > 0 . y > 0 . Since y = 100 − 2 x , y = 100 − 2 x , if y > 0 , y > 0 , then x < 50 . x < 50 . Therefore, we are trying to determine the maximum value of A ( x ) A ( x ) for x x over the open interval ( 0 , 50 ) . ( 0 , 50 ) . We do not know that a function necessarily has a maximum value over an open interval. However, we do know that a continuous function has an absolute maximum (and absolute minimum) over a closed interval. Therefore, let’s consider the function A ( x ) = 100 x − 2 x 2 A ( x ) = 100 x − 2 x 2 over the closed interval [ 0 , 50 ] . [ 0 , 50 ] . If the maximum value occurs at an interior point, then we have found the value x x in the open interval ( 0 , 50 ) ( 0 , 50 ) that maximizes the area of the garden. Therefore, we consider the following problem:

Maximize A ( x ) = 100 x − 2 x 2 A ( x ) = 100 x − 2 x 2 over the interval [ 0 , 50 ] . [ 0 , 50 ] .

As mentioned earlier, since A A is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum. These extreme values occur either at endpoints or critical points. At the endpoints, A ( x ) = 0 . A ( x ) = 0 . Since the area is positive for all x x in the open interval ( 0 , 50 ) , ( 0 , 50 ) , the maximum must occur at a critical point. Differentiating the function A ( x ) , A ( x ) , we obtain

Therefore, the only critical point is x = 25 x = 25 ( Figure 4.63 ). We conclude that the maximum area must occur when x = 25 . x = 25 . Then we have y = 100 − 2 x = 100 − 2 ( 25 ) = 50 . y = 100 − 2 x = 100 − 2 ( 25 ) = 50 . To maximize the area of the garden, let x = 25 x = 25 ft and y = 50 ft . y = 50 ft . The area of this garden is 1250 ft 2 . 1250 ft 2 .

Checkpoint 4.31

Determine the maximum area if we want to make the same rectangular garden as in Figure 4.63 , but we have 200 200 ft of fencing.

Now let’s look at a general strategy for solving optimization problems similar to Example 4.32 .

Problem-Solving Strategy

Problem-solving strategy: solving optimization problems.

  • Introduce all variables. If applicable, draw a figure and label all variables.
  • Determine which quantity is to be maximized or minimized, and for what range of values of the other variables (if this can be determined at this time).
  • Write a formula for the quantity to be maximized or minimized in terms of the variables. This formula may involve more than one variable.
  • Write any equations relating the independent variables in the formula from step 3 . 3 . Use these equations to write the quantity to be maximized or minimized as a function of one variable.
  • Identify the domain of consideration for the function in step 4 4 based on the physical problem to be solved.
  • Locate the maximum or minimum value of the function from step 4 . 4 . This step typically involves looking for critical points and evaluating a function at endpoints.

Now let’s apply this strategy to maximize the volume of an open-top box given a constraint on the amount of material to be used.

Example 4.33

Maximizing the volume of a box.

An open-top box is to be made from a 24 24 in. by 36 36 in. piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. What size square should be cut out of each corner to get a box with the maximum volume?

Step 1: Let x x be the side length of the square to be removed from each corner ( Figure 4.64 ). Then, the remaining four flaps can be folded up to form an open-top box. Let V V be the volume of the resulting box.

Step 2: We are trying to maximize the volume of a box. Therefore, the problem is to maximize V . V .

Step 3: As mentioned in step 2 , 2 , we are trying to maximize the volume of a box. The volume of a box is V = L · W · H , V = L · W · H , where L , W , and H L , W , and H are the length, width, and height, respectively.

Step 4: From Figure 4.64 , we see that the height of the box is x x inches, the length is 36 − 2 x 36 − 2 x inches, and the width is 24 − 2 x 24 − 2 x inches. Therefore, the volume of the box is

Step 5: To determine the domain of consideration, let’s examine Figure 4.64 . Certainly, we need x > 0 . x > 0 . Furthermore, the side length of the square cannot be greater than or equal to half the length of the shorter side, 24 24 in.; otherwise, one of the flaps would be completely cut off. Therefore, we are trying to determine whether there is a maximum volume of the box for x x over the open interval ( 0 , 12 ) . ( 0 , 12 ) . Since V V is a continuous function over the closed interval [ 0 , 12 ] , [ 0 , 12 ] , we know V V will have an absolute maximum over the closed interval. Therefore, we consider V V over the closed interval [ 0 , 12 ] [ 0 , 12 ] and check whether the absolute maximum occurs at an interior point.

Step 6: Since V ( x ) V ( x ) is a continuous function over the closed, bounded interval [ 0 , 12 ] , [ 0 , 12 ] , V V must have an absolute maximum (and an absolute minimum). Since V ( x ) = 0 V ( x ) = 0 at the endpoints and V ( x ) > 0 V ( x ) > 0 for 0 < x < 12 , 0 < x < 12 , the maximum must occur at a critical point. The derivative is

To find the critical points, we need to solve the equation

Dividing both sides of this equation by 12 , 12 , the problem simplifies to solving the equation

Using the quadratic formula, we find that the critical points are

Since 10 + 2 7 10 + 2 7 is not in the domain of consideration, the only critical point we need to consider is 10 − 2 7 . 10 − 2 7 . Therefore, the volume is maximized if we let x = 10 − 2 7 in . x = 10 − 2 7 in . The maximum volume is V ( 10 − 2 7 ) = 640 + 448 7 ≈ 1825 in . 3 V ( 10 − 2 7 ) = 640 + 448 7 ≈ 1825 in . 3 as shown in the following graph.

Watch a video about optimizing the volume of a box.

Checkpoint 4.32

Suppose the dimensions of the cardboard in Example 4.33 are 20 in. by 30 in. Let x x be the side length of each square and write the volume of the open-top box as a function of x . x . Determine the domain of consideration for x . x .

Example 4.34

Minimizing travel time.

An island is 2 mi 2 mi due north of its closest point along a straight shoreline. A visitor is staying at a cabin on the shore that is 6 mi 6 mi west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of 8 mph 8 mph and swims at a rate of 3 mph . 3 mph . How far should the visitor run before swimming to minimize the time it takes to reach the island?

Step 1: Let x x be the distance running and let y y be the distance swimming ( Figure 4.66 ). Let T T be the time it takes to get from the cabin to the island.

Step 2: The problem is to minimize T . T .

Step 3: To find the time spent traveling from the cabin to the island, add the time spent running and the time spent swimming. Since Distance = = Rate × × Time ( D = R × T ) , ( D = R × T ) , the time spent running is

and the time spent swimming is

Therefore, the total time spent traveling is

Step 4: From Figure 4.66 , the line segment of y y miles forms the hypotenuse of a right triangle with legs of length 2 mi 2 mi and 6 − x mi . 6 − x mi . Therefore, by the Pythagorean theorem, 2 2 + ( 6 − x ) 2 = y 2 , 2 2 + ( 6 − x ) 2 = y 2 , and we obtain y = ( 6 − x ) 2 + 4 . y = ( 6 − x ) 2 + 4 . Thus, the total time spent traveling is given by the function

Step 5: From Figure 4.66 , we see that 0 ≤ x ≤ 6 . 0 ≤ x ≤ 6 . Therefore, [ 0 , 6 ] [ 0 , 6 ] is the domain of consideration.

Step 6: Since T ( x ) T ( x ) is a continuous function over a closed, bounded interval, it has a maximum and a minimum. Let’s begin by looking for any critical points of T T over the interval [ 0 , 6 ] . [ 0 , 6 ] . The derivative is

If T ′ ( x ) = 0 , T ′ ( x ) = 0 , then

Squaring both sides of this equation, we see that if x x satisfies this equation, then x x must satisfy

which implies

We conclude that if x x is a critical point, then x x satisfies

Therefore, the possibilities for critical points are

Since x = 6 + 6 / 55 x = 6 + 6 / 55 is not in the domain, it is not a possibility for a critical point. On the other hand, x = 6 − 6 / 55 x = 6 − 6 / 55 is in the domain. Since we squared both sides of Equation 4.6 to arrive at the possible critical points, it remains to verify that x = 6 − 6 / 55 x = 6 − 6 / 55 satisfies Equation 4.6 . Since x = 6 − 6 / 55 x = 6 − 6 / 55 does satisfy that equation, we conclude that x = 6 − 6 / 55 x = 6 − 6 / 55 is a critical point, and it is the only one. To justify that the time is minimized for this value of x , x , we just need to check the values of T ( x ) T ( x ) at the endpoints x = 0 x = 0 and x = 6 , x = 6 , and compare them with the value of T ( x ) T ( x ) at the critical point x = 6 − 6 / 55 . x = 6 − 6 / 55 . We find that T ( 0 ) ≈ 2.108 h T ( 0 ) ≈ 2.108 h and T ( 6 ) ≈ 1.417 h, T ( 6 ) ≈ 1.417 h, whereas T ( 6 − 6 / 55 ) ≈ 1.368 h . T ( 6 − 6 / 55 ) ≈ 1.368 h . Therefore, we conclude that T T has a local minimum at x ≈ 5.19 x ≈ 5.19 mi.

Checkpoint 4.33

Suppose the island is 1 1 mi from shore, and the distance from the cabin to the point on the shore closest to the island is 15 mi . 15 mi . Suppose a visitor swims at the rate of 2.5 mph 2.5 mph and runs at a rate of 6 mph . 6 mph . Let x x denote the distance the visitor will run before swimming, and find a function for the time it takes the visitor to get from the cabin to the island.

In business, companies are interested in maximizing revenue . In the following example, we consider a scenario in which a company has collected data on how many cars it is able to lease, depending on the price it charges its customers to rent a car. Let’s use these data to determine the price the company should charge to maximize the amount of money it brings in.

Example 4.35

Maximizing revenue.

Owners of a car rental company have determined that if they charge customers p p dollars per day to rent a car, where 50 ≤ p ≤ 200 , 50 ≤ p ≤ 200 , the number of cars n n they rent per day can be modeled by the linear function n ( p ) = 1000 − 5 p . n ( p ) = 1000 − 5 p . If they charge $ 50 $ 50 per day or less, they will rent all their cars. If they charge $ 200 $ 200 per day or more, they will not rent any cars. Assuming the owners plan to charge customers between $50 per day and $ 200 $ 200 per day to rent a car, how much should they charge to maximize their revenue?

Step 1: Let p p be the price charged per car per day and let n n be the number of cars rented per day. Let R R be the revenue per day.

Step 2: The problem is to maximize R . R .

Step 3: The revenue (per day) is equal to the number of cars rented per day times the price charged per car per day—that is, R = n × p . R = n × p .

Step 4: Since the number of cars rented per day is modeled by the linear function n ( p ) = 1000 − 5 p , n ( p ) = 1000 − 5 p , the revenue R R can be represented by the function

Step 5: Since the owners plan to charge between $ 50 $ 50 per car per day and $ 200 $ 200 per car per day, the problem is to find the maximum revenue R ( p ) R ( p ) for p p in the closed interval [ 50 , 200 ] . [ 50 , 200 ] .

Step 6: Since R R is a continuous function over the closed, bounded interval [ 50 , 200 ] , [ 50 , 200 ] , it has an absolute maximum (and an absolute minimum) in that interval. To find the maximum value, look for critical points. The derivative is R ′ ( p ) = −10 p + 1000 . R ′ ( p ) = −10 p + 1000 . Therefore, the critical point is p = 100 p = 100 When p = 100 , p = 100 , R ( 100 ) = $ 50,000 . R ( 100 ) = $ 50,000 . When p = 50 , p = 50 , R ( p ) = $ 37,500 . R ( p ) = $ 37,500 . When p = 200 , p = 200 , R ( p ) = $ 0 . R ( p ) = $ 0 . Therefore, the absolute maximum occurs at p = $ 100 . p = $ 100 . The car rental company should charge $ 100 $ 100 per day per car to maximize revenue as shown in the following figure.

Checkpoint 4.34

A car rental company charges its customers p p dollars per day, where 60 ≤ p ≤ 150 . 60 ≤ p ≤ 150 . It has found that the number of cars rented per day can be modeled by the linear function n ( p ) = 750 − 5 p . n ( p ) = 750 − 5 p . How much should the company charge each customer to maximize revenue?

Example 4.36

Maximizing the area of an inscribed rectangle.

A rectangle is to be inscribed in the ellipse

What should the dimensions of the rectangle be to maximize its area? What is the maximum area?

Step 1: For a rectangle to be inscribed in the ellipse, the sides of the rectangle must be parallel to the axes. Let L L be the length of the rectangle and W W be its width. Let A A be the area of the rectangle.

Step 2: The problem is to maximize A . A .

Step 3: The area of the rectangle is A = L W . A = L W .

Step 4: Let ( x , y ) ( x , y ) be the corner of the rectangle that lies in the first quadrant, as shown in Figure 4.68 . We can write length L = 2 x L = 2 x and width W = 2 y . W = 2 y . Since x 2 4 + y 2 = 1 x 2 4 + y 2 = 1 and y > 0 , y > 0 , we have y = 1 − x 2 4 . y = 1 − x 2 4 . Therefore, the area is

Step 5: From Figure 4.68 , we see that to inscribe a rectangle in the ellipse, the x x -coordinate of the corner in the first quadrant must satisfy 0 < x < 2 . 0 < x < 2 . Therefore, the problem reduces to looking for the maximum value of A ( x ) A ( x ) over the open interval ( 0 , 2 ) . ( 0 , 2 ) . Since A ( x ) A ( x ) will have an absolute maximum (and absolute minimum) over the closed interval [ 0 , 2 ] , [ 0 , 2 ] , we consider A ( x ) = 2 x 4 − x 2 A ( x ) = 2 x 4 − x 2 over the interval [ 0 , 2 ] . [ 0 , 2 ] . If the absolute maximum occurs at an interior point, then we have found an absolute maximum in the open interval.

Step 6: As mentioned earlier, A ( x ) A ( x ) is a continuous function over the closed, bounded interval [ 0 , 2 ] . [ 0 , 2 ] . Therefore, it has an absolute maximum (and absolute minimum). At the endpoints x = 0 x = 0 and x = 2 , x = 2 , A ( x ) = 0 . A ( x ) = 0 . For 0 < x < 2 , 0 < x < 2 , A ( x ) > 0 . A ( x ) > 0 . Therefore, the maximum must occur at a critical point. Taking the derivative of A ( x ) , A ( x ) , we obtain

To find critical points, we need to find where A ′ ( x ) = 0 . A ′ ( x ) = 0 . We can see that if x x is a solution of

then x x must satisfy

Therefore, x 2 = 2 . x 2 = 2 . Thus, x = ± 2 x = ± 2 are the possible solutions of Equation 4.7 . Since we are considering x x over the interval [ 0 , 2 ] , [ 0 , 2 ] , x = 2 x = 2 is a possibility for a critical point, but x = − 2 x = − 2 is not. Therefore, we check whether 2 2 is a solution of Equation 4.7 . Since x = 2 x = 2 is a solution of Equation 4.7 , we conclude that 2 2 is the only critical point of A ( x ) A ( x ) in the interval [ 0 , 2 ] . [ 0 , 2 ] . Therefore, A ( x ) A ( x ) must have an absolute maximum at the critical point x = 2 . x = 2 . To determine the dimensions of the rectangle, we need to find the length L L and the width W . W . If x = 2 x = 2 then

Therefore, the dimensions of the rectangle are L = 2 x = 2 2 L = 2 x = 2 2 and W = 2 y = 2 2 = 2 . W = 2 y = 2 2 = 2 . The area of this rectangle is A = L W = ( 2 2 ) ( 2 ) = 4 . A = L W = ( 2 2 ) ( 2 ) = 4 .

Checkpoint 4.35

Modify the area function A A if the rectangle is to be inscribed in the unit circle x 2 + y 2 = 1 . x 2 + y 2 = 1 . What is the domain of consideration?

Solving Optimization Problems when the Interval Is Not Closed or Is Unbounded

In the previous examples, we considered functions on closed, bounded domains. Consequently, by the extreme value theorem, we were guaranteed that the functions had absolute extrema. Let’s now consider functions for which the domain is neither closed nor bounded.

Many functions still have at least one absolute extrema, even if the domain is not closed or the domain is unbounded. For example, the function f ( x ) = x 2 + 4 f ( x ) = x 2 + 4 over ( − ∞ , ∞ ) ( − ∞ , ∞ ) has an absolute minimum of 4 4 at x = 0 . x = 0 . Therefore, we can still consider functions over unbounded domains or open intervals and determine whether they have any absolute extrema. In the next example, we try to minimize a function over an unbounded domain. We will see that, although the domain of consideration is ( 0 , ∞ ) , ( 0 , ∞ ) , the function has an absolute minimum.

In the following example, we look at constructing a box of least surface area with a prescribed volume. It is not difficult to show that for a closed-top box, by symmetry, among all boxes with a specified volume, a cube will have the smallest surface area. Consequently, we consider the modified problem of determining which open-topped box with a specified volume has the smallest surface area.

Example 4.37

Minimizing surface area.

A rectangular box with a square base, an open top, and a volume of 216 216 in. 3 is to be constructed. What should the dimensions of the box be to minimize the surface area of the box? What is the minimum surface area?

Step 1: Draw a rectangular box and introduce the variable x x to represent the length of each side of the square base; let y y represent the height of the box. Let S S denote the surface area of the open-top box.

Step 2: We need to minimize the surface area. Therefore, we need to minimize S . S .

Step 3: Since the box has an open top, we need only determine the area of the four vertical sides and the base. The area of each of the four vertical sides is x · y . x · y . The area of the base is x 2 . x 2 . Therefore, the surface area of the box is

Step 4: Since the volume of this box is x 2 y x 2 y and the volume is given as 216 in . 3 , 216 in . 3 , the constraint equation is

Solving the constraint equation for y , y , we have y = 216 x 2 . y = 216 x 2 . Therefore, we can write the surface area as a function of x x only:

Therefore, S ( x ) = 864 x + x 2 . S ( x ) = 864 x + x 2 .

Step 5: Since we are requiring that x 2 y = 216 , x 2 y = 216 , we cannot have x = 0 . x = 0 . Therefore, we need x > 0 . x > 0 . On the other hand, x x is allowed to have any positive value. Note that as x x becomes large, the height of the box y y becomes correspondingly small so that x 2 y = 216 . x 2 y = 216 . Similarly, as x x becomes small, the height of the box becomes correspondingly large. We conclude that the domain is the open, unbounded interval ( 0 , ∞ ) . ( 0 , ∞ ) . Note that, unlike the previous examples, we cannot reduce our problem to looking for an absolute maximum or absolute minimum over a closed, bounded interval. However, in the next step, we discover why this function must have an absolute minimum over the interval ( 0 , ∞ ) . ( 0 , ∞ ) .

Step 6: Note that as x → 0 + , x → 0 + , S ( x ) → ∞ . S ( x ) → ∞ . Also, as x → ∞ , x → ∞ , S ( x ) → ∞ . S ( x ) → ∞ . Since S S is a continuous function that approaches infinity at the ends, it must have an absolute minimum at some x ∈ ( 0 , ∞ ) . x ∈ ( 0 , ∞ ) . This minimum must occur at a critical point of S . S . The derivative is

Therefore, S ′ ( x ) = 0 S ′ ( x ) = 0 when 2 x = 864 x 2 . 2 x = 864 x 2 . Solving this equation for x , x , we obtain x 3 = 432 , x 3 = 432 , so x = 432 3 = 6 2 3 . x = 432 3 = 6 2 3 . Since this is the only critical point of S , S , the absolute minimum must occur at x = 6 2 3 x = 6 2 3 (see Figure 4.70 ). When x = 6 2 3 , x = 6 2 3 , y = 216 ( 6 2 3 ) 2 = 3 2 3 in . y = 216 ( 6 2 3 ) 2 = 3 2 3 in . Therefore, the dimensions of the box should be x = 6 2 3 in . x = 6 2 3 in . and y = 3 2 3 in . y = 3 2 3 in . With these dimensions, the surface area is

Checkpoint 4.36

Consider the same open-top box, which is to have volume 216 in . 3 . 216 in . 3 . Suppose the cost of the material for the base is 20 ¢ / in . 2 20 ¢ / in . 2 and the cost of the material for the sides is 30 ¢ / in . 2 30 ¢ / in . 2 and we are trying to minimize the cost of this box. Write the cost as a function of the side lengths of the base. (Let x x be the side length of the base and y y be the height of the box.)

For the following exercises, answer by proof, counterexample, or explanation.

When you find the maximum for an optimization problem, why do you need to check the sign of the derivative around the critical points?

Why do you need to check the endpoints for optimization problems?

True or False . For every continuous nonlinear function, you can find the value x x that maximizes the function.

True or False . For every continuous nonconstant function on a closed, finite domain, there exists at least one x x that minimizes or maximizes the function.

For the following exercises, set up and evaluate each optimization problem.

To carry a suitcase on an airplane, the length + width + + width + height of the box must be less than or equal to 62 in . 62 in . Assuming the base of the suitcase is square, show that the volume is V = h ( 31 − ( 1 2 ) h ) 2 . V = h ( 31 − ( 1 2 ) h ) 2 . What height allows you to have the largest volume?

You are constructing a cardboard box with the dimensions 2 m by 4 m . 2 m by 4 m . You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions of the box with the largest volume?

Find the positive integer that minimizes the sum of the number and its reciprocal.

Find two positive integers such that their sum is 10 , 10 , and minimize and maximize the sum of their squares.

For the following exercises, consider the construction of a pen to enclose an area.

You have 400 ft 400 ft of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?

You have 800 ft 800 ft of fencing to make a pen for hogs. If you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area?

You need to construct a fence around an area of 1600 ft 2 . 1600 ft 2 . What are the dimensions of the rectangular pen to minimize the amount of material needed?

Two poles are connected by a wire that is also connected to the ground. The first pole is 20 ft 20 ft tall and the second pole is 10 ft 10 ft tall. There is a distance of 30 ft 30 ft between the two poles. Where should the wire be anchored to the ground to minimize the amount of wire needed?

[T] You are moving into a new apartment and notice there is a corner where the hallway narrows from 8 ft to 6 ft . 8 ft to 6 ft . What is the length of the longest item that can be carried horizontally around the corner?

A patient’s pulse measures 70 bpm, 80 bpm, then 120 bpm . 70 bpm, 80 bpm, then 120 bpm . To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression ( x − 70 ) 2 + ( x − 80 ) 2 + ( x − 120 ) 2 ? ( x − 70 ) 2 + ( x − 80 ) 2 + ( x − 120 ) 2 ? What value minimizes it?

In the previous problem, assume the patient was nervous during the third measurement, so we only weight that value half as much as the others. What is the value that minimizes ( x − 70 ) 2 + ( x − 80 ) 2 + 1 2 ( x − 120 ) 2 ? ( x − 70 ) 2 + ( x − 80 ) 2 + 1 2 ( x − 120 ) 2 ?

You can run at a speed of 6 6 mph and swim at a speed of 3 3 mph and are located on the shore, 4 4 miles east of an island that is 1 1 mile north of the shoreline. How far should you run west to minimize the time needed to reach the island?

For the following problems, consider a lifeguard at a circular pool with diameter 40 m . 40 m . He must reach someone who is drowning on the exact opposite side of the pool, at position C . C . The lifeguard swims with a speed v v and runs around the pool at speed w = 3 v . w = 3 v .

Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, θ . θ .

Find at what angle θ θ the lifeguard should swim to reach the drowning person in the least amount of time.

A truck uses gas as g ( v ) = a v + b v , g ( v ) = a v + b v , where v v represents the speed of the truck and g g represents the gallons of fuel per mile. Assuming a a and b b are positive, at what speed is fuel consumption minimized?

For the following exercises, consider a limousine that gets m ( v ) = ( 120 − 2 v ) 5 mi/gal m ( v ) = ( 120 − 2 v ) 5 mi/gal at speed v , v , the chauffeur costs $15/h , $15/h , and gas is $ 3.5 / gal . $ 3.5 / gal .

Find the cost per mile at speed v . v .

Find the cheapest driving speed.

For the following exercises, consider a pizzeria that sell pizzas for a revenue of R ( x ) = a x R ( x ) = a x and costs C ( x ) = b + c x + d x 2 , C ( x ) = b + c x + d x 2 , where x x represents the number of pizzas ;   a   >   c ;   a   >   c .

Find the profit function for the number of pizzas. How many pizzas gives the largest profit per pizza?

Assume that R ( x ) = 10 x R ( x ) = 10 x and C ( x ) = 2 x + x 2 . C ( x ) = 2 x + x 2 . How many pizzas sold maximizes the profit?

Assume that R ( x ) = 15 x , R ( x ) = 15 x , and C ( x ) = 60 + 3 x + 1 2 x 2 . C ( x ) = 60 + 3 x + 1 2 x 2 . How many pizzas sold maximizes the profit?

For the following exercises, consider a wire 4 ft 4 ft long cut into two pieces. One piece forms a circle with radius r r and the other forms a square of side x . x .

Choose x x to maximize the sum of their areas.

Choose x x to minimize the sum of their areas.

For the following exercises, consider two nonnegative numbers x x and y y such that x + y = 10 . x + y = 10 . Maximize and minimize the quantities.

x 2 y 2 x 2 y 2

y − 1 x y − 1 x

x 2 − y x 2 − y

For the following exercises, draw the given optimization problem and solve.

Find the volume of the largest right circular cylinder that fits in a sphere of radius 1 . 1 .

Find the volume of the largest right cone that fits in a sphere of radius 1 . 1 .

Find the area of the largest rectangle that fits into the triangle with sides x = 0 , y = 0 x = 0 , y = 0 and x 4 + y 6 = 1 . x 4 + y 6 = 1 .

Find the largest volume of a cylinder that fits into a cone that has base radius R R and height h . h .

Find the dimensions of the closed cylinder volume V = 16 π V = 16 π that has the least amount of surface area.

Find the dimensions of a right cone with surface area S = 4 π S = 4 π that has the largest volume.

For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.

[T] Where is the line y = 5 − 2 x y = 5 − 2 x closest to the origin?

[T] Where is the line y = 5 − 2 x y = 5 − 2 x closest to point ( 1 , 1 ) ? ( 1 , 1 ) ?

[T] Where is the parabola y = x 2 y = x 2 closest to point ( 2 , 0 ) ? ( 2 , 0 ) ?

[T] Where is the parabola y = x 2 y = x 2 closest to point ( 0 , 3 ) ? ( 0 , 3 ) ?

For the following exercises, set up, but do not evaluate, each optimization problem.

A window is composed of a semicircle placed on top of a rectangle. If you have 20 ft 20 ft of window-framing materials for the outer frame, what is the maximum size of the window you can create? Use r r to represent the radius of the semicircle.

You have a garden row of 20 20 watermelon plants that produce an average of 30 30 watermelons apiece. For any additional watermelon plants planted, the output per watermelon plant drops by one watermelon. How many extra watermelon plants should you plant?

You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $ 5 / ft 2 $ 5 / ft 2 and the material for the sides costs $ 2 / ft 2 . $ 2 / ft 2 . You need a box with volume 4 ft 3 . 4 ft 3 . Find the dimensions of the box that minimize cost. Use x x to represent the length of the side of the box.

You are building five identical pens adjacent to each other with a total area of 1000 m 2 , 1000 m 2 , as shown in the following figure. What dimensions should you use to minimize the amount of fencing?

You are the manager of an apartment complex with 50 50 units. When you set rent at $ 800 / month, $ 800 / month, all apartments are rented. As you increase rent by $ 25 / month, $ 25 / month, one fewer apartment is rented. Maintenance costs run $ 50 / month $ 50 / month for each occupied unit. What is the rent that maximizes the total amount of profit?

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Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction
  • Authors: Gilbert Strang, Edwin “Jed” Herman
  • Publisher/website: OpenStax
  • Book title: Calculus Volume 1
  • Publication date: Mar 30, 2016
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/calculus-volume-1/pages/1-introduction
  • Section URL: https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems

© Feb 5, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

PRECALCULUS OPTIMIZATION PROBLEMS WITH SOLUTIONS

Steps for solving optimization problems.

1) Read the problem.

2) Sketch a picture if possible and use variables for unknown quantities.

3) Write a function, expressing the quantity to be maximized or minimized as a function of one or more variables.

4) If your function has more than one variable, use information from the rest of the problem to solve for this variable in terms of the other

5) Determine the domain of the independent variable (the values for which the stated problem makes sense.)

6) Determine the maximum and minimum values by using your graphing calculator. Draw a sketch of the function you used, label your answer on your sketch, and then write your answer in a sentence.

Example Problems of Optimization

Example 1 :

An open box is to be made from a rectangular piece of cardstock, 8.5 inches wide and 11 inches tall, by cutting out squares of equal size from the four corners and bending up the sides. Find the maximum volume that the box can have. What size squares should be cut to create the box of maximum volume?

Let x be the side length of square to cut out.

precalculus worksheet on optimization answers

Volume of the box  =  length  ⋅ width  ⋅ height

  =    (8.5 - 2x)  ⋅ (11 - 2x)  ⋅ x

Domain of the function :

8.5 - 2x  =  0

2x  =  8.5

x  =  4.25

Domain is 0 < x < 4.25

precalculus worksheet on optimization answers

Maximum value is 66.148. So the side length of square to cut out is 1.585 inches.

Example 2 :

A farmer has 120 feet of fencing with which to enclose two adjacent rectangular pens as shown. What dimensions should be used so that the enclosed area will be a maximum? What will the area be?

precalculus worksheet on optimization answers

Let x and y be the length and width of the rectangular field respectively.

Length of fencing  =  120 feet

x + y + x + y + y  =  120

2x + 3y  =  120

3y  =  120 - 2x

y  =  (120 - 2x)/3

Area of the filed  =  length (width)

  =  x  ⋅  [ (120 - 2x)/3]

0 < x < 60

Using graphing calculator, we get

precalculus worksheet on optimization answers

Maximum at x  =  30

Applying the value of y, we get

y  =  (120 - 60)/3

y  =  60/3

y  =  20

Area of the rectangular field  =  30(20)

  =  600 square feet

So, we get the maximum area enclosed by the dimension 30 feet and 20 feet. The maximum area is 600 square feet.

Example 3 :

A closed box with a square base must have a volume of 5000 cu. cm. Find the dimensions of the box that will minimize the amount of material used.

Volume of the box with a square base  =  5000 cu.cm

precalculus worksheet on optimization answers

Let x be the length and width of the square base of cuboid and y be the height of cuboid.

Volume of cuboid  = length  ⋅ width  ⋅ height

  =  x  ⋅ x  ⋅ y

  =  x 2 y

x 2  y  =  5000

y  =  5000/ x 2

In order to find the material to be minimized, we should find the surface area of cuboid.

Surface area of closed cuboid box  =  2(lb + bh + hl)

  =  2(x 2 + xy + xy)

  =  2(x 2  + 2xy)

  =   2x 2  + 4xy

  =  2x 2  + 4x( 5000/ x 2 )

S(x)  =   2x 2  + (20 000/ x )

The volume of cuboid created by applying the material will not exceed 5000.

Domain is 0 < x <  √5000

Using graphing calculator, we find

precalculus worksheet on optimization answers

Side length of square base  =  17.1 cm

length of cuboid  =  17.1

Breadth of cuboid  =  17.1

Height of cuboid  ( y)  =  5000/( 17.1) 2

  =  5000/292.41

  =  17.1

Hence the dimensions are 17.1 cm and 17.1 cm.

precalculus worksheet on optimization answers

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Math 122B - First Semester Calculus and 125 - Calculus I Worksheets

The following is a list of worksheets and other materials related to Math 122B and 125 at the UA. Your instructor might use some of these in class. You may also use any of these materials for practice. The chapter headings refer to Calculus , Sixth Edition by Hughes-Hallett et al. Published by Wiley.

INTRODUCTION

  • Tools for Success -A list of resources including tutoring services. website
  • Student Survey - A survey to provide background information to an instructor.  pdf   doc
  • Calculator Checklist - A list of calculator skills that are required for Calculus.  pdf   doc
  • Pixels and the calculator screen - An exercise to illustrate the sensitivity of the window settings.  pdf   doc
  • Homework Sample - A few examples to illustrate how homework should be written.  pdf   doc

CHAPTER 1 - A Library of Functions

  • Interesting Graphs - A few equations to graph that have interesting (and hidden) features.  pdf   doc
  • Functions - Properties of functions and the Rule of Four (equations, tables, graphs, and words).  pdf   doc
  • Reading a Position Graph - Answer questions about motion using a position graph.  pdf   doc
  • Reading Graphs - Four graphs and questions using function notation.  pdf   doc
  • Find a Function - Find an example of a function in the media.  pdf   doc
  • INDY 500 - Sketch graphs based on traveling one lap along an oval racetrack.  pdf   doc
  • Farenheit - The relationship between Farenheit and Celsius.  pdf   doc
  • Linear Functions - Applications.  pdf   doc
  • Exponential Functions - Recognizing exponential functions and their properties.  pdf   doc
  • Inverse Functions - Relationships between a function and its inverse.  pdf   doc
  • New Functions From Old - Transformations, compositions, and inverses of functions.  pdf   doc
  • Transformations - A matching exercise using symbolic expressions and tables.  pdf   doc
  • More Transformations - Graphing transformation.   pdf   doc
  • Logarithms - Using logarithms to solve problems. Properties of logas.  pdf   doc
  • Trig Reference Sheet - List of basic identities and rules.   pdf   doc
  • Trig (part I) -Interpreting trig functions and practice with inverses.  pdf   doc
  • Trig (part II) - More practice.  pdf   doc
  • Denise & Chad - An illustration of the effects of changes in amplitude and period.  pdf   doc
  • Polynomials & Rational Functions - Recognizing polynomials and rational functions and their properties.  pdf   doc
  • Power Functions - Use graphs to explore power functions.  pdf   doc
  • Limits and Continuity - Graphical and numerical exercises.  pdf   doc
  • More Continuity - Basics about continuity.  pdf   doc

CHAPTER 2 - The Derivative

  • Introduction to Rates - Introduction to rates of change using position and velocity.   pdf   doc
  • Representations - Symbolic recognition and illustration of rates. Practical interpretation of rates of change using the rule of four.   pdf   doc
  • Practical Example - Reading information about rates from a graph.  pdf   doc
  • Estimation - Estimation using tables and equations. Practice with notation and terminology.  pdf   doc
  • Derivative Graphs - Graphing a derivative function given a graph.  pdf   doc
  • More Derivative Graphs - Matching exercise.  pdf   doc
  • Terminology - Fill in the blank exercise. Practice with terminology  pdf   doc
  • Differentiability - Determine when a function is not differentiable at a point.  pdf   doc
  • More Differentiability - More practice.   pdf   doc
  • Practice - Additional practice covering this section.  pdf   doc

CHAPTER 3 - Rules For Differentiation

  • Product & Quotient Rules - Practice using these rules.   pdf   doc
  • Chain Rule - Practice using this rule.   pdf   doc
  • Base e - Derivation of e using derivatives.  pdf   doc
  • Rules - Practice with tables and derivative rules in symbolic form.  pdf   doc
  • More Practice - More practice using all the derivative rules.  pdf   doc
  • Derivative (&Integral) Rules - A table of derivative and integral rules.   pdf   doc

CHAPTER 4 - Using the Derivative

  • Reading Graphs - Reading information from first and second derivative graphs.  pdf   doc
  • Critical Points Part I - Terminology and characteristics of critical points.  pdf   doc
  • Critical Points Part II - Finding critical points and graphing.  pdf   doc
  • Families of Functions - Finding critical points for families of functions.  pdf   doc
  • More Families of Functions - Finding values of parameters in families of functions.  pdf   doc
  • Optimization Part I - Optimization problems emphasizing geometry.  pdf   doc
  • Optimization Part II - More optimization problems.  pdf   doc
  • Parametric Equations (Circles) - Sketching variations of the standard parametric equations for the unit circle.  pdf   doc
  • Parametric Equations (Misc) - Fun graphs using parametric equations.   pdf   doc
  • Parametric Equations - Finding direction of motion and tangent lines using parametric equations.  pdf   doc
  • Holiday Parametric Equations - Halloween surprise.   pdf   doc
  • L'Hopital's Rule - Practice in recognizing when to use L'Hopital's Rule.  pdf   doc
  • Limit Practice -Additional practice with limits including L'Hopital's Rule.   pdf   doc
  • Introduction to Related Rates - Finding various derivatives using volume of a sphere and surface area of a cylinder.  pdf   doc
  • Related Rates - Additional practice.  pdf   doc
  • More Related Rates -Additional practice.   pdf   doc

CHAPTER 5 - The Definite Integral

  • Intro to Velocity and Area - Relationship between velocity, position, and area.  pdf   doc
  • Representations - Practice with notation, estimation, and interpretations.  pdf   doc
  • Rocket - Application of velocity and position for a model rocket.  pdf   doc
  • Mice - Application of velocity and position for two mice.  pdf   doc
  • Cars - Application of velocity, position, and acceleration of two cars.  pdf   doc
  • Fundamental Theorem Part I - Graphical approach.  pdf   doc
  • Fundamental Theorem Part II - Illustrations and notation.  pdf   doc

CHAPTER 6 - Constructing Antiderivatives

  • Position, Velocity, & Acceleration - Graphical relationships between position, velocity, and acceleration.  pdf   doc
  • Sketching Antiderivatives - Graphing antiderivatives.   pdf   doc
  • Area Between Graphs - Using the Fundamental Theorem to find area between graphs.  pdf   doc
  • Practice - Problems from chapters 5 and 6.  pdf   doc
  • Integration - Recognizing when to use substitution. Integrands look similar.  pdf   doc
  • Substitution - Practice, including definite integrals.  pdf   doc
  • More Substitution - More practice.  pdf   doc

IMAGES

  1. Optimization Worksheet with Answer Key by The Teach U Shop

    precalculus worksheet on optimization answers

  2. Answer Key Precalculus Worksheets With Answers / 7 Best Images of

    precalculus worksheet on optimization answers

  3. Answer Key Precalculus Worksheets With Answers / Measuring Angles

    precalculus worksheet on optimization answers

  4. Honors Precalculus Worksheets / Pre Calculus Honors Mrs Higgins Calc

    precalculus worksheet on optimization answers

  5. Precalculus Worksheets With Answers Pdf

    precalculus worksheet on optimization answers

  6. cpm precalculus answers chapter 2

    precalculus worksheet on optimization answers

VIDEO

  1. Precalculus Worksheet 2 Section 2.6

  2. Optimization Problem 5

  3. MATH1061 Optimization Worksheet #1

  4. Precalculus

  5. Optimization Problems Part 1 (Precalculus 1)

  6. Verifying Trig Equations Using Trig Identities

COMMENTS

  1. PDF Calc

    1. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum. 2. Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum. 3.

  2. PDF Math 120: Precalculus Autumn 2014

    Step 1. Write a formula for the quantity you want to optimize. If you don't know how, then you probably need to draw a better picture, or label more things in your picture. Step 2. (If your formula involves only one variable, you can skip this part and move on to step 3.)

  3. 4.7 Applied Optimization Problems

    Problem-Solving Strategy Problem-Solving Strategy: Solving Optimization Problems Introduce all variables. If applicable, draw a figure and label all variables. Determine which quantity is to be maximized or minimized, and for what range of values of the other variables (if this can be determined at this time).

  4. PDF Nov 13

    Precalculus Honors Optimization Worksheet #3 l. 16 feet of wire is used to a square and a circle. Find the length of the sides of the square and the radius enclose the minimum total area. I a-xo 2. A horse breeder wants to construct a corral next to a horse barn 50 feet long, using the bam as one side of the corral.

  5. PDF Worksheet 24: Optimization

    100 P( 2) = 3 And thus I = 2 is the unique maximum. p 2. Find the point on the curve y = x that is closest to the point (3; 0). Note that we wish to minimize the distance between the given function and the given point; we use, therefore, the distance formula: d = p(x 3)2 + (y 0)2 p

  6. Calculus I

    Solution Here is a set of practice problems to accompany the Optimization section of the Applications of Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University.

  7. PDF Optimization Date Period

    Optimization Name___________________________________ Date________________ Period____ Solve each optimization problem. You may use the provided box to sketch the problem setup and the provided graph to sketch the function of one variable to be minimized or maximized.

  8. PRECALCULUS OPTIMIZATION PROBLEMS WITH SOLUTIONS

    Solution : Let x be the side length of square to cut out. Volume of the box = length ⋅ width ⋅ height = (8.5 - 2x) ⋅ (11 - 2x) ⋅ x Domain of the function : 8.5 - 2x = 0 2x = 8.5 x = 4.25 Domain is 0 < x < 4.25 Maximum value is 66.148. So the side length of square to cut out is 1.585 inches. Example 2 :

  9. Precalculus

    Find step-by-step solutions and answers to Precalculus - 9780076602186, as well as thousands of textbooks so you can move forward with confidence. ... Preparing for Precalculus. Section 0-1: Sets. Section 0-2: Operations with Complex Numbers. ... Linear Optimization. Page 413: Study Guide and Review. Page 417: Chapter Test. Exercise 1. Exercise ...

  10. PDF PRECALCULUS WORKSHEET ON OPTIMIZATION Steps for Solving Optimization

    PRECALCULUS WORKSHEET ON OPTIMIZATION Steps for Solving Optimization Problems: 1) Read the problem. 2) Sketch a picture if possible. Label the picture, using variables for unknown quantities. 3) Write a function, expressing the quantity to be maximized or minimized as a function of one or more variables.

  11. PDF AMAT100: Pre-Calculus Worksheet 2

    AMAT100: Pre-Calculus Worksheet 2 Due: 2/14 (MW Sections) and 2/15 (TTh Sections) in Class or Digitally Name: UAlbany Email: Instructions • This homework should be submitted in class or digitally on the date listed above. • There are three main ways you might want to write up your work. - Write on this pdf using a tablet

  12. PDF 5.11 Solving Optimization Problems Practice Calculus

    Calculus 1. A particle is traveling along the -axis and it's position from the origin can be modeled by 12 1 where is meters and is minutes on the interval . a. At what time during the interval 0 4 is the particle farthest to the left? Practice b. On the same interval what is the particle's maximum speed? 2.

  13. PDF Optimization Problems Practice

    Answers to Optimization Problems Practice p = the profit per day x = the number of items manufactured per day Function to maximize: p = x( 110 − 0.05 x) − ( 50 x + 6000) where 0 ≤ x < ∞ Optimal number of smartphones to manufacture per day: 600 A = the total area of the two corrals x = the length of the non-adjacent sides of each corral

  14. Free Printable Math Worksheets for Precalculus

    Definition of the derivative. Instantaneous rates of change. Power rule for differentiation. Motion along a line. Approximating area under a curve. Area under a curve by limit of sums. Indefinite integrals. Free Precalculus worksheets created with Infinite Precalculus. Printable in convenient PDF format.

  15. Optimization (practice)

    Google Classroom An open-topped glass aquarium with a square base is designed to hold 62.5 cubic feet of water. What is the minimum possible exterior surface area of the aquarium? square feet Stuck? Report a problem Do 4 problems

  16. PDF Calculus Practice: Optimization 1

    Smallest product of the two numbers: . . 2) A supermarket employee wants to construct an open-top box from a by in piece of cardboard. To do this, the employee plans to cut out squares of. V = the volume of the box x = the length of the sides of the squares Function to maximize: V ( x)( x) x where x .

  17. PDF AB Calculus

    Answers to Optimization Practice A = the total area of the two corrals x = the length of the non-adjacent sides of each corral 100 - 4 = 2 x x Function to maximize: where 0 < x < 25 3 25 50 Dimensions of each corall: 2 ft (non-adjecent sides) by ft (adjacent 3 sides)

  18. Math 180 Calculus 1 Worksheets

    This booklet contains worksheets for the Math 180 Calculus 1 course at the University of Illinois at Chicago. There are 27 worksheets, each covering a certain topic of the course curriculum. At the end of the booklet there are 2 review worksheets, covering parts of the course (based on a two-midterm model). In a 15-week semester, completing 2 ...

  19. Math 124/125

    Math 122B - First Semester Calculus and 125 - Calculus I. Worksheets. The following is a list of worksheets and other materials related to Math 122B and 125 at the UA. Your instructor might use some of these in class. You may also use any of these materials for practice. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et ...

  20. PDF Calculus: Optimization Worksheet Name:

    Calculus: Optimization Worksheet Name: _____ 3. Four pens will be built side by side along a wall by using 150 feet of fencing. What dimensions will maximize the area of the pens. 4. Suppose you had to use exactly 200 m of fencing to make either one square enclosure or two separate square enclosures of any size you wished.

  21. Calculus Worksheets

    These Calculus Worksheets will produce word problems that deal with the optimization of resources in scenarios. The student will be given a function and will be asked to list the points at which that the tangent line to that function is horizontal. You may select the number of problems. These Optimization Worksheets are a great resource for ...

  22. PDF Calc

    1. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum. 2. Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum.

  23. Quiz & Worksheet

    This worksheet and quiz let you practice the following skills: Critical thinking - apply relevant concepts to examine information about optimization problems in calculus in a different light ...

  24. PDF WS 03.6 Optimization

    Calculator permitted. Show all set-ups and analysis. Report all answers to 3 decimals and avoid intermediate rounding error. Multiple Choice An advertisement is run to stimulate the sale of cars. After t days, 1 ≤ t ≤ 48 , the number of cars sold is given by N ( t ) = 4000 + 45 t 2 − t 3 . On what day does the maximum rate of growth sales occur?