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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.