Exam 4: Applications of Differentiation

The graph of the derivative of $f ^ { \prime } ( x )$ of a function f(x) is given below. $The graph of the derivative of f ^ { \prime } ( x ) of a function f(x) is given below. Sketch a possible graph of y = f ( x ) on the interval [ - 2,2 ] such that f ( - 2 ) = 0 . On your graph, label the critical point(s) and inflection point(s).$ Sketch a possible graph of $y = f ( x )$ on the interval $[ - 2,2 ]$ such that $f ( - 2 ) = 0$ . On your graph, label the critical point(s) and inflection point(s).

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Question 201

Sketch the graph of a function f that is continuous on [0,4] and has the following properties: Critical numbers at 1 and 3, a local minimum at 1, but no local maximum or local minimum at 3.

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Question 202

Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time. When the height is 5 in. and is increasing at 0.2 in./s, the radius is 3 in. and is decreasing at 0.1 in./s. How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant?

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Question 203

In its report to stockholders, a certain manufacturing company reported its total cost and revenue functions using the graphs given below: $In its report to stockholders, a certain manufacturing company reported its total cost and revenue functions using the graphs given below: (a) Using the graph, sketch each of the following as a function of the quantity produced, x.(i) Marginal cost (ii) Marginal revenue (iii) Profit (b) What is the significance of x _ { 1 } ? x _ { 2 } ? (c) What is the meaning of C ^ { \prime } ( 0 ) ? (d) Describe in words how the marginal cost changes as the quantity produced increases.(e) Explain the economic significance, in terms of marginal cost, of the point where the concavity of the cost function changes.$ (a) Using the graph, sketch each of the following as a function of the quantity produced, x.(i) Marginal cost (ii) Marginal revenue (iii) Profit (b) What is the significance of $x _ { 1 }$ ? $x _ { 2 }$ ? (c) What is the meaning of $C ^ { \prime } ( 0 )$ ? (d) Describe in words how the marginal cost changes as the quantity produced increases.(e) Explain the economic significance, in terms of marginal cost, of the point where the concavity of the cost function changes.

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Question 204

On the domain $( 0 , \infty )$ , find the minimum value of $f ( x ) = \frac { e ^ { x } } { x }$ .

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Question 205

A stone falls from a cliff and reaches the ground at a speed 180 miles per hour. What is the height of the cliff above the ground? (Assume there is no air resistance.)

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Question 206

If $f ^ { \prime \prime } ( x ) = 12 x ^ { 2 } - 2 , f ^ { \prime } ( - 1 ) = - 4$ , and $f ( 0 ) = 2$ , find $f ( 1 )$ .

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Question 207

If f is continuous over the closed interval $[ 1,4 ]$ and differentiable over the open interval $( 1,4 )$ , and that $f ^ { \prime } ( x ) \leq 5$ for all x in $[ 1,4 ]$ . If $f ( 1 ) = 2$ , what is the maximum possible value that f(4) can be?

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Question 208

A boat floating several feet away from a dock is pulled toward the dock by a rope wound by a winch at the rate of 20 feet per minute. If the winch is 10 feet above the level of the boat, how fast is the boat moving through the water when there is 100 feet of rope between the boat and the winch?

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Question 209

A ladder 10 feet long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at 3 feet per second, how fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 feet away from the wall?

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Question 210

Let $g ( x ) = x \sqrt { 1 - x }$ . Use the graphs of g and $g ^ { \prime }$ to estimate, to two decimal places, (a) the critical numbers of g.(b) the interval(s) where g is increasing.(c) the interval(s) where g is decreasing..

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Question 211

A 400-meter-long running track is to be constructed with two parallel straightaways of length s joining two semi-circular curves of radius r. What are the dimensions of the track that will enclose the largest infield area?

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Question 212

A poster is to have a total area of 486 square inches with 3-inch margins at the bottom and top and a 2-inch margin at each side. What dimensions of a poster will give the largest printed area?

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Question 213

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { e ^ { x ^ { 2 } } - 1 } { x }$ .

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Question 214

Is it true that $\int \sqrt { 4 - x ^ { 2 } } d x = \frac { x } { 2 } \sqrt { 4 - x ^ { 2 } } + \arcsin \left( \frac { x } { 2 } \right) + C$ ? Explain why or why not.

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Question 215

A square is to be cut from each corner of a piece of paper which is 8 cm by 10 cm, and the sides are to be folded up to create an open box. What should the side of the square be for maximum volume? (State your answer correct to two decimal places.)

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Question 216

Find the most general antiderivative of the function $f ( x ) = 9 x ^ { 2 }$ .

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Question 217

Determine the largest interval on which the function $f ( x ) = x + \frac { 1 } { x }$ concave upward?

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Question 218

Find the absolute maximum values of $f ( x ) = x ^ { 3 } - 3 x$ over the set of all real numbers x such that $x ^ { 2 } \leq 3 x$

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Question 219

A cylindrical can is to be made with volume $V _ { 0 }$ . Find dimensions that will minimize the amount of material used in its construction. (Assume the thickness of the material is uniform.)

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Question 220

Showing 201 - 220 of 273

Sketch the graph of a function f that is continuous on [0,4] and has the following properties: Critical numbers at 1 and 3, a local minimum at 1, but no local maximum or local minimum at 3.

On the domain $( 0 , \infty )$ , find the minimum value of $f ( x ) = \frac { e ^ { x } } { x }$ .

A stone falls from a cliff and reaches the ground at a speed 180 miles per hour. What is the height of the cliff above the ground? (Assume there is no air resistance.)

If $f ^ { \prime \prime } ( x ) = 12 x ^ { 2 } - 2 , f ^ { \prime } ( - 1 ) = - 4$ , and $f ( 0 ) = 2$ , find $f ( 1 )$ .

If f is continuous over the closed interval $[ 1,4 ]$ and differentiable over the open interval $( 1,4 )$ , and that $f ^ { \prime } ( x ) \leq 5$ for all x in $[ 1,4 ]$ . If $f ( 1 ) = 2$ , what is the maximum possible value that f(4) can be?

A boat floating several feet away from a dock is pulled toward the dock by a rope wound by a winch at the rate of 20 feet per minute. If the winch is 10 feet above the level of the boat, how fast is the boat moving through the water when there is 100 feet of rope between the boat and the winch?

A ladder 10 feet long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at 3 feet per second, how fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 feet away from the wall?

Let $g ( x ) = x \sqrt { 1 - x }$ . Use the graphs of g and $g ^ { \prime }$ to estimate, to two decimal places, (a) the critical numbers of g.(b) the interval(s) where g is increasing.(c) the interval(s) where g is decreasing..

A 400-meter-long running track is to be constructed with two parallel straightaways of length s joining two semi-circular curves of radius r. What are the dimensions of the track that will enclose the largest infield area?

A poster is to have a total area of 486 square inches with 3-inch margins at the bottom and top and a 2-inch margin at each side. What dimensions of a poster will give the largest printed area?

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { e ^ { x ^ { 2 } } - 1 } { x }$ .

Is it true that $\int \sqrt { 4 - x ^ { 2 } } d x = \frac { x } { 2 } \sqrt { 4 - x ^ { 2 } } + \arcsin \left( \frac { x } { 2 } \right) + C$ ? Explain why or why not.

A square is to be cut from each corner of a piece of paper which is 8 cm by 10 cm, and the sides are to be folded up to create an open box. What should the side of the square be for maximum volume? (State your answer correct to two decimal places.)

Find the most general antiderivative of the function $f ( x ) = 9 x ^ { 2 }$ .

Determine the largest interval on which the function $f ( x ) = x + \frac { 1 } { x }$ concave upward?

Find the absolute maximum values of $f ( x ) = x ^ { 3 } - 3 x$ over the set of all real numbers x such that $x ^ { 2 } \leq 3 x$

A cylindrical can is to be made with volume $V _ { 0 }$ . Find dimensions that will minimize the amount of material used in its construction. (Assume the thickness of the material is uniform.)

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Limits and Derivatives

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Exam 4: Applications of Differentiation

Sketch the graph of a function f that is continuous on [0,4] and has the following properties: Critical numbers at 1 and 3, a local minimum at 1, but no local maximum or local minimum at 3.

On the domain (0,∞)( 0 , \infty )(0,∞) , find the minimum value of f(x)=exxf ( x ) = \frac { e ^ { x } } { x }f(x)=xex​ .

A stone falls from a cliff and reaches the ground at a speed 180 miles per hour. What is the height of the cliff above the ground? (Assume there is no air resistance.)

If f′′(x)=12x2−2,f′(−1)=−4f ^ { \prime \prime } ( x ) = 12 x ^ { 2 } - 2 , f ^ { \prime } ( - 1 ) = - 4f′′(x)=12x2−2,f′(−1)=−4 , and f(0)=2f ( 0 ) = 2f(0)=2 , find f(1)f ( 1 )f(1) .

If f is continuous over the closed interval [1,4][ 1,4 ][1,4] and differentiable over the open interval (1,4)( 1,4 )(1,4) , and that f′(x)≤5f ^ { \prime } ( x ) \leq 5f′(x)≤5 for all x in [1,4][ 1,4 ][1,4] . If f(1)=2f ( 1 ) = 2f(1)=2 , what is the maximum possible value that f(4) can be?

A boat floating several feet away from a dock is pulled toward the dock by a rope wound by a winch at the rate of 20 feet per minute. If the winch is 10 feet above the level of the boat, how fast is the boat moving through the water when there is 100 feet of rope between the boat and the winch?

A ladder 10 feet long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at 3 feet per second, how fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 feet away from the wall?

Let g(x)=x1−xg ( x ) = x \sqrt { 1 - x }g(x)=x1−x​ . Use the graphs of g and g′g ^ { \prime }g′ to estimate, to two decimal places, (a) the critical numbers of g.(b) the interval(s) where g is increasing.(c) the interval(s) where g is decreasing..

A 400-meter-long running track is to be constructed with two parallel straightaways of length s joining two semi-circular curves of radius r. What are the dimensions of the track that will enclose the largest infield area?

A poster is to have a total area of 486 square inches with 3-inch margins at the bottom and top and a 2-inch margin at each side. What dimensions of a poster will give the largest printed area?

Find the value of the limit lim⁡x→0+ex2−1x\lim _ { x \rightarrow 0 ^ { + } } \frac { e ^ { x ^ { 2 } } - 1 } { x }limx→0+​xex2−1​ .

Is it true that ∫4−x2dx=x24−x2+arcsin⁡(x2)+C\int \sqrt { 4 - x ^ { 2 } } d x = \frac { x } { 2 } \sqrt { 4 - x ^ { 2 } } + \arcsin \left( \frac { x } { 2 } \right) + C∫4−x2​dx=2x​4−x2​+arcsin(2x​)+C ? Explain why or why not.

A square is to be cut from each corner of a piece of paper which is 8 cm by 10 cm, and the sides are to be folded up to create an open box. What should the side of the square be for maximum volume? (State your answer correct to two decimal places.)

Find the most general antiderivative of the function f(x)=9x2f ( x ) = 9 x ^ { 2 }f(x)=9x2 .

Determine the largest interval on which the function f(x)=x+1xf ( x ) = x + \frac { 1 } { x }f(x)=x+x1​ concave upward?

Find the absolute maximum values of f(x)=x3−3xf ( x ) = x ^ { 3 } - 3 xf(x)=x3−3x over the set of all real numbers x such that x2≤3xx ^ { 2 } \leq 3 xx2≤3x

A cylindrical can is to be made with volume V0V _ { 0 }V0​ . Find dimensions that will minimize the amount of material used in its construction. (Assume the thickness of the material is uniform.)

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

On the domain $( 0 , \infty )$ , find the minimum value of $f ( x ) = \frac { e ^ { x } } { x }$ .

If $f ^ { \prime \prime } ( x ) = 12 x ^ { 2 } - 2 , f ^ { \prime } ( - 1 ) = - 4$ , and $f ( 0 ) = 2$ , find $f ( 1 )$ .

If f is continuous over the closed interval $[ 1,4 ]$ and differentiable over the open interval $( 1,4 )$ , and that $f ^ { \prime } ( x ) \leq 5$ for all x in $[ 1,4 ]$ . If $f ( 1 ) = 2$ , what is the maximum possible value that f(4) can be?

Let $g ( x ) = x \sqrt { 1 - x }$ . Use the graphs of g and $g ^ { \prime }$ to estimate, to two decimal places, (a) the critical numbers of g.(b) the interval(s) where g is increasing.(c) the interval(s) where g is decreasing..

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { e ^ { x ^ { 2 } } - 1 } { x }$ .

Is it true that $\int \sqrt { 4 - x ^ { 2 } } d x = \frac { x } { 2 } \sqrt { 4 - x ^ { 2 } } + \arcsin \left( \frac { x } { 2 } \right) + C$ ? Explain why or why not.

Find the most general antiderivative of the function $f ( x ) = 9 x ^ { 2 }$ .

Determine the largest interval on which the function $f ( x ) = x + \frac { 1 } { x }$ concave upward?

Find the absolute maximum values of $f ( x ) = x ^ { 3 } - 3 x$ over the set of all real numbers x such that $x ^ { 2 } \leq 3 x$

A cylindrical can is to be made with volume $V _ { 0 }$ . Find dimensions that will minimize the amount of material used in its construction. (Assume the thickness of the material is uniform.)