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Calculus I                                                                      Test #2                                             November 3, 2004

Name____________________                               R.  Hammack                                             Score ______
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(1) Suppose f(x) is a function for which the following limits hold.

Based on this information, answer the following questions.

(a) Is there a value of x at which the tangent line to the graph of y = f(x) is horizontal?  If so, what is x?
The third limit says  f '(2) = 0, so x = 2 is such a value.

(b) Find the slope of the tangent line to the graph of  y = f(x)  at the point where x = 5.
The fourth limit says  f '(5) = 3, so the slope is 3.

(c)  Suppose you also know that f(0) = 5.  Find the equation of the tangent line to the graph of   y = f(x)  at the point where x = 0.
The first limit says  f '(0) = -1, so the tangent line has slope -1 and y-intercept 5. It's equation is thus y = -x +5

(2) The graph of a function g(x) is illustrated below.  Sketch the graph of g '(x).

(3) If ,  find

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)   Find the values of x at which the tangent to the graph of     is horizontal.

If the tangent is horizontal, its slope   must be 0, so we need to solve

Answer: x = 3 and x = 1

(14)  An object, moving on a straight line, is at a distance of    feet from a stationary point on the line at time t seconds.

(a)  Find the function for the object's velocity at time t.

Velocity at time t is feet per second

(b)  At what time(s) is the object's velocity 5 feet per second?

When f '(t) = 5.

Answer: when t = 3 seconds and t = 5 seconds

(15)  At which values of x do the graphs of    and    have the same slope?

Answer: x = 0 and x = 2/3

(16)   Find the slope of the tangent to the graph of the equation   at the point (3, 2).

(17)  A stone dropped into a still pond sends out a circular ripple whose radius increases at a rate of 2 feet per second.  How rapidly is the area enclosed by the ripple increasing 10 seconds after the stone is dropped?

Let r be the radius of the ripple, and let A be its area.

We know

We want ?

r and A obey the equation   (area of a circle).  Differentiating both sides with respect to time t,

10 seconds after the stone is dropped, the ripple's radius is (10 sec)(2 ft/sec) = 20 feet.
Plugging this in, we get square feet per second.