Calculus Test 2 Outline and Practice
You should be able to:
1. Use First principles to determine a derivative.
2. Differentiates expressions including the use of product, quotient and chain rules.
3. Differentiate expressions when they are written either explicitly or implicitly.
4. Determine the equation of either the normal or tangent lines to a function.
5. Use the first and second derivatives to sketch a graph
6. Interpret the derivative
7. Simplify derivatives.
8. Evaluate derivatives
9. Apply the derivative to application problems given a function.
10. Solve related rates questions
Practice Questions
1. To study traffic flow along a speedway, a city installs a device in front of a city property. The device
counts the number of cars driving by and records the total periodically. The resulting data is plotted on
a graph with time, in hours, on the horizontal axis and the total number of cars since 4am, in thousands
on the vertical axis. The graph of C(t) is shown below.
a) When is the traffic flow greatest?
C ′(3)
C ′(3) ?
b) From the graph, estimate
c) What is the meaning of
d) What are its units?
e) ? What does the value you obtained in
in practical terms
2.
part b) mean
Use first principles to determine the derivative for each of the following functions.
a)
y = 3x 2 − 5 x + 1
b) y =
3 − 2x
x+5
c)
y = 2 − 3x
3.
Given the function: y = 2 x 3 + 6 x 2 − 48 x − 4 . Use calculus to determine the x coordinates where
this function would have its min and max values.
4.
Given that f ( x) = 
2 x − 5, x < −1

 . Determine the values of a and b necessary if f(x) is
2
ax − 3 x + b, x ≥ −1
differentiable.
g ( x) = 5 x 2 − 3x + 2
5.
For the function
6.
The temperature, T, in degrees Fahrenheit, of a cold yam placed in a hot oven is given by T(t) ,
where t is the time in minutes since the yam was put in the oven.
i)
What is the sign of T ′(t ) ? Explain.
7.
find the equation of the normal line at x = 2.
T ′(20) ?
ii)
What are the units of
iii)
What is the practical meaning of the statement
T ′(20) = 2 ?
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Calculus Test 2 Outline and Practice
8.
Air is being pumped into a spherical balloon at a rate of 5 cm3/min. Determine the rate at which
the radius of the balloon is increasing when the diameter of the balloon is 20 cm.
9.
A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and
is being pushed towards the wall at a rate of ft/sec. How fast is the top of the ladder moving up
the wall 12 seconds after we start pushing?
10.
. The base
A tank of water in the shape of a cone is leaking water at a constant rate of
radius of the tank is 5 ft and the height of the tank is 14 ft. At what rate is the depth of the water in
the tank changing when the depth of the water is 6 ft?
11.
A spherical snowball melts in such a way that the instant at which its radius is 20 cm, its radius is
decreasing at 3 cm/min. At what rate is the volume of the ball of snow changing at that instant?
13.
A ladder 25 feet long is leaning against the wall of the house, while the base of the ladder is pulled
away from the wall at a rate of 3 feet per second. How fast is the top moving down the wall when
the base of the ladder is 20 feet away from the wall?
14.
A kite is flying 150 m high, where the wind causes it to move horizontally at the rate of 5 m per
second. In order to maintain the kite at a height of 150 m , the person must allow more string to be
let out. At what rate is the string being let out when the length of the string already out is 250 m?
15.
Water runs into an inverted conical tank at the rate of 7 cubic feet per minute. The radius of the
water’s surface is always half the height of the water.
a) How fast is the water level rising when the water is 2 feet deep?
b) Suppose water is leaking out of the tank at a rate of 2 feet/sec in (a). How fast is the radius of
surface of the water increasing when the water is 2 feet deep?
16.
Determine the interval(s) for which the function:
g ( x) = x 4 / 3 (3 x + 6) is:
a) increasing
b) concave up
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