# AP Calculus TEST #3 Outline Emphasis on Integral Calculus

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"AP Calculus TEST #3 Outline Emphasis on Integral Calculus"

```					AP Calculus TEST #3 Outline
Emphasis on Integral Calculus
You should be able to:

1.    Determine the derivative of a function using the definition of a derivative (or first
principles).
2.    Differentiate various functions (constant to a variable, variable to a variable,
variable to a constant, trigonometric, inverse trigonometric, logarithmic and
implicit)
3.    Sketch the derivative of a function.
4.    Sketch a function and label critical points (including min, max and inflection
points).
5.    Use the second derivative test to determine if a critical point is a min or a max.
6.    Determine absolute and relative (or local) min or max values for a function.
7.    Solve related rate and optimization problems of various types
8.    Use differentials to solve problems.
9.    Discuss the continuity and differentiability of a function.
10.   Use linearization to determine a specific value and be able to explain how this
process works.
11.   Determine the limit of a function algebraically, numerically or graphically.
12.   Determine the equation of a normal or tangent line to a curve at a given point.
13.   Calculate the average or the instantaneous growth of a function.
14.   Determine the area under a curve from a table of values or a graph and the use of
Riemann Sums.
15.   Use technology to determine the value of a derivative or integral
16.   Interpret derivatives and integrals.
17.   Determine the value of an integral (simple ones only) by hand
18.   Understand Rolle’s Theorem, the Mean Valve Theorem for derivatives and the
Mean Value theorem for Integrals and be able to apply them.
19.   Understand and be able to apply the Fundamental Theorem of Calculus (Parts 1
and 2)
20.   Solve simple first order differential equations involving motion.

Some Practice Questions:

1     The table below shows the amount of water leaking out of a tank at a rate of r(t)
litres per hour. Give an upper and lower estimate of the total quantity of water
that has leaked out of the tank in 12 hours.

Time          1   2   3   4   5   6   7   8   9     10    11    12
(hours)
Leakage       8   6   5   5   4   3   2   1   0.5   0.5   0.3   0.2
(litres/hour)
2.    Given the velocity, in km per hour, of a car can be modeled by the function:
v(t ) = 2t 2 + −4t + 10 . Determine the distance it has traveled during the time
interval t ∈ [0,8] using four rectangles and MRAM . Illustrate your answer on
the diagram.

with a diagram.

3

∫
7                            0

a)
−1
5 ∂x                       b)    ∫ (5 − x) ∂x
−2
c)   ∫
−5
25 − x 2 ∂x

4
⎧2 x, x < 2 ⎫
d)    ∫ f ( x) ∂x ; given that:
0
f ( x) = ⎨
⎩4, x ≥ 2⎭
⎬

4      After investing \$1000 at an annual interest rate of 7% compounded continuously
for t years, you balance is \$B, where B = f(t).

i)          What are the units of          ∂B    ?
∂t

ii)         What is the financial interpretation of        ∂B   ?
∂t
5   Let P(t) represent the price of a share of stock of a corporation at time t. What
does each of the following statements tell us about the signs of the first and
second derivatives of P(t)?
i) “The price of the stock is rising faster and faster.”
ii) “The price of the stock is close to bottoming out.”

6   The rate of depreciation for a new piece of equipment at a factory is given as
p(t ) = 50t − 600 for t ∈ [0,10] , where t is measured in years. Write an integral
that would represent the total loss of value of the equipment over the first five
years and then evaluate the interval.

7   If R(t) is the profit that a company has made, measured in dollars per month, and t
is measured in months (0 representing April 1st; t =1 representing May 1 etc.),
6

what does        ∫ R(t ) ∂t = 5600
0
mean?

8   Water is leaking out of a barrel at a rate of r (t ) litres per minute. Write a definite
integral expressing the total quantity of water which leaks out of the barrel in the
first 2 hours.

9   Use part 1 of the Fundamental Theorem to compute each integral exactly.
Determine the following integrals by hand. Answers should be exact.

2                                            3

∫ (2 x − 3) ∂x                               ∫ (x       − 2) ∂x
2
a)                                           b)
0                                            0

4                                            2
2
c)   ∫ ( x + 3x) ∂x
0
d)   ∫ (4 x − x
1
2
) ∂x

π                                            π
2                                            4
e)   ∫ (2 sin x) ∂x
0
f)   ∫ (sec x tan x) ∂x
0

1                                            3

∫ (e                                         ∫ (3e
−x
g)          x
− e ) ∂x                     h)              2x
− x 2 ) ∂x
0                                            0

x −3
4                                            5
6
i)   ∫ x ∂x
1
j)   ∫ 3x + 4 ∂x
2
10   Suppose that, for a particular population of organisms, the birth rate is given by
b(t ) = 410 − 0.3t organisms per month and the death rate is given by
12
d (t ) = 390 + 0.2t organisms per month. Determine ∫ [b(t ) − d (t )] ∂t and explain
0

what it represents.

11   Suppose that the temperature t months into the year is given
π
by: T (t ) = 64 − 24 cos( t ) in degrees Fahrenheit. Estimate the average
6
temperature over an entire year. Explain why this answer is obvious from the
graph of T(t).

12   The linear density of a rod of length 4 m is given by ρ ( x) = 9 + 2 x measured in
kilograms per meter, where x is measured in meters from one end of the rod. Find
the total mass of the rod.

x
1
13   Find the interval on which the curve y = ∫                       ∂t is concave upward.
0 1+ t + t
2

3x                    5x
1
14   Find the derivatives of :     ∫x u + 1 ∂u and
2
∫ cos(2u ) ∂u
cos x

t2
1+ u4
x
15   If F ( x) = ∫ f (t ) ∂t , where f (t ) =   ∫          ∂u , find F ′′(2)
1                          1
u

16   Given the graph of p(x) as shown below, determine h(3), h’(3) and h” (3) if
x
h( x ) =   ∫ p(t ) ∂t
−1
x
17   Solve for x given that: x>0 and: ∫ (t 3 − 2t + 3) ∂t = 4 .
0

x
18   Find the linearlization of: h( x) = 3 + ∫ 5 cos(2t ) ∂t at x = 0
0

x
19   Find f(3) if   ∫ f (t ) ∂t = 5
0
x +1

20   A dam released 1000 m3 of water at 10m3/min and then another 1000 m3 at
20m3/min. What was the average rate at which the water was released? Give

21   A car is 130 m from a stop sign when it begins to decelerate at a rate of   t m/s2.
If its speed when it begins to decelerate is 20m/s:
a) when will it come to a full stop?
b) how far from the stop sign will it be when it stops?

22   You are driving along a highway at a steady 60mph (88ft/sec) when you see an
accident ahead and slam on the brakes. What constant deceleration is required to
stop your car in 242 feet?

23   Suppose that the marginal cost of manufacturing an item when x thousand units
∂c
are produced is:    = 3x 2 − 12 x + 15 dollars per item. Find the cost function
∂x
c(x) if c(0) = 400.

24   The acceleration of a particle moving along a coordinate line is 2 + 6t m/s2. The
initial velocity (at time zero) is 4m/s.
a) Find the velocity as a function of time t.
b) How far does the particle move during the first second of its trip?

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 views: 24 posted: 7/29/2010 language: English pages: 5