CALCULUS DICE QUIZ NAME:_________________ CHAPTER 5 DATE: _________________

SHOW ALL WORK !!!

NO GRAPHING CALCULATORS !!!!!

1. Let g and h be any two twice-differentiable functions that are defined for all real numbers and that satisfy the following properties for all x:

(I)

!

g(x)( )2

+ h(x)( )2

=1

(II)

!

g'(x) = h(x)( )2

(III)

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h(x) > 0 (IV)

!

g(0) = 0 A. Justify that

!

h'(x) = "g(x)h(x) for all x. B. Justify that h has a relative maximum at x = 0. C. Justify that the graph of g has a point of inflection at x = 0.

2. Consider the function f defined by

!

f (x) = x2"1( )

3

for all real numbers x.

A. For what values of x is the function increasing? B. Find the x- and y-coordinates of the relative maximum and minimum

points. Justify your answer. C. For what values of x is the graph of f concave upward? D. Using the information found in part A, B, and C, sketch the graph of f on

the axis provided.

3. Let h be a function defined for all

!

x " 0 such that h(4) = -3 and the derivative of h

is given by

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h'(x) =x2" 2

x for all

!

x " 0 .

A. Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answer.

B. On what intervals, if any, is the graph of h concave up? Justify your answer.

C. Write an equation for the line tangent to the graph of h at x = 4. D. Does the line tangent to the graph of h at x = 4 lie above or below the

graph of h for x > 4. Why?

4. Suppose that the function f has a continuous second derivative for all x, and that f(0) = 2, f’(0) = -3, and f”(0) = 0. Let g be a function whose derivative is given by

!

g'(x) = e"2x(3 f (x) + 2 f '(x)) for all x.

A. Write an equation of the line tangent to the graph of f at the point where x = 0.

B. Is there sufficient information to determine whether or nor the graph of f has a point of inflection when x = 0? Explain your answer.

C. Given that g(0) = 4, write an equation of the line tangent to the graph of g at the point where x = 0.

D. Show that

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g"(x) = e"2x("6 f (x) " f '(x) + 2 f "(x)) . Does g have a local

maximum at x = 0? Justify your answer.

5. Let f be the function given by

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f (x) = 2xe2x .

A. Find

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limx"#$

f (x) and

!

limx"+#

f (x). B. Find the absolute maximum value of f. Justify that your answer is an

absolute minimum. C. What is the range of f? D. Consider the family of functions defined by

!

y = bxebx , where b is a

nonzero constant. Show that the absolute minimum value of

!

bxebx is the

same for all nonzero values of b.

6. The figure below shows the graph of f’, the derivative of the function f, for

!

"7 # x # 7 . The graph of f’ has horizontal tangent lines at x = -3, x = 2, and x = 5 and vertical tangent line at x = 3.

A. Find all values of x for

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"7 # x # 7 , at which f attains a relative maximum. Justify your answer.

B. Find all values of x for

!

"7 # x # 7 , at which f”(x) < 0. C. At what value of x for

!

"7 # x # 7 , does f attain an absolute maximum.? Justify your answer.

D. Find all values of x, for

!

"7 # x # 7 , at which f attains a relative minimum. Justify your answer.

f’(x)

-7 -5 -1 3 5

-1

7. A particle moves along the x-axis so that at time t its position is given by

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x(t) = sin("t 2) for

!

"1# t #1. A. Find the velocity at time t. B. Find the acceleration at time t. C. For what values of t does the particle change direction? D. Find all values of t for which the particle is moving to the left.

8. A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and volume is 36 cubic meters. If building the tank costs $10 per square meter for the base and $5 per square meter for the sides, what is the cost of the least expensive tank?

9. A man has 340 yards of fencing for enclosing two separate fields, one of which is to be a rectangle twice as long as is wide and the other a square. The square field must contain at least 800 square yards.

A. If x is the width of the rectangular field, what are the maximum and minimum possible values of x?

B. What is the greatest number of square yards that can be enclosed in the two fields? Justify your answer.

10. A manufacturer finds it costs him x

!

x2

+ 5x + 7 dollars to produce x tons of an item. At production levels above 3 tons, he must hire additional workers, and his costs increase by

!

3(x " 3) dollars on his total production. If the price he receives is $13 per ton regardless of how much he manufactures and if he has a plant capacity of 10 tons, what level of output maximizes his profits?

11. Find the area of the largest rectangle( with sides parallel to the coordinate axes) that can be inscribed in the region enclosed by the graphs of

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f (x) =18 " x2 and

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g(x) = 2x2" 9

12. A rectangle ABCD with sides parallel to the coordinate axes is inscribed in the region enclosed by the graph of

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y = "4x2

+ 4 and the x-axis as shown in the figure below. A. Find the x- and y-coordinates of C so that the area of the rectangle ABCD

is a maximum. B. The point C moves along the curve with its x-coordinate increasing at the

constant rate of 2 units per second. Find the rate of change of the area of rectangle ABCD when x = ½

!

y = 4x2

+ 4

A B

C D