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Calculus
Semester 1
Review
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Show all work!
1. The graph of f is given. (chapter 1)
a) Find the value of f(-1)
b) Find the Domain of f
c) Find the Range of f
d) Estimate the values of x such that f(x)=0
e) On what interval is f decreasing?
2. For the function f whose graph is given, state the value of the given quantity, if it
exists. If it does not exist, explain why. (chapter 2)
(a) limx0 f ( x)
(b) limx3 f ( x)
(c) limx3 f ( x)
(d) limx3 f ( x)
3. Evaluate each limit, if it exists. (chapter 2)
(a) lim x2
x2  x  6
x2
(b) lim x2
x2  x  6
x2
(c) lim x3
x2  9
2x2  7x  3
(d) limh0
(4  h) 2  16
h
4. Use the definition of a derivative to find the derivative of f. (Show your work)
(chapter 3)
f(x) = 3 – 2x + 4x2
5. Find the derivative of each function. (chapter 3)
(a) f(x) = 152
(b) f(x) =x2 – 3x + 4
(c) y  x ( x  1)
(d) y 
3x  1
2x  1
(e) y = x-3sinx
(f) y 
1  sin x
x  cos x
(g) y  ( x 3  1)100
(h) y  x 3 cos(nx)
6. Find y  by implicit differentiation. (with respect to x) (chapter 3.7)
(a) x 2  y 2  10
(b) y 5  x 2 y 3  1  x 4 y
7. Find the first and second derivatives of the function. (chapter 3.8)
(a) y  x 8  7 x 6  2 x 4
(b) y  x sin x
8. If a ball is given a push so that it has an initial velocity of 5 m/s down a certain inclined
plane, then the distance it has rolled after t seconds is s = 5t + 3t2. (chapter 3)
(a) Find the velocity after 2 seconds.
(b) How long does it take for the velocity to reach 35 m/s?
9. The cost function for a certain commodity is C(x) = 84 +0.16x – 0.0006x2 + .000003x3
Find C/(100) and explain what it means. (chapter 3)
10. A ladder 15 feet long is leaning against a building. The foot of the ladder is being
moved away from the building at a constant rate of ½ foot per second. (chapter 3.9)
(a) Find the rate at which the top of the ladder is moving down the building when the foot
of the ladder is 9 feet from the building.
(b) Find the rate of change of the area of the triangle formed when the foot of the ladder
is 9 feet from the building.
11. The equation of motion is given for a particle, where s is in meters and t is in seconds.
S= 2t3 – 3t2 – 12t
(chapter 3)
(a) find the velocity function:
(b) find the acceleration function:
(c) find the acceleration after 1 second:
(d) find the acceleration at the instants when velocity is 0:
12. A plane flying horizontally at an altitude of 1 mile and a speed of 500 miles per hour
passes directly over a radar station. Find the rate at which the distance from the plane to
the station is increasing when it is 2 miles away from the station. (chapter 3.9)
13. Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = 2x3 – 3x2 – 12x +1, [-2, 3]
(chapter 4)
14. Use the Mean Value Theorem to find the value of c if f(x) = x3 – 2x2 +3x – 2 over the
interval [0, 2]. (chapter 4.2)
15.For f(x) = 5 – 3x2 +x3 find:
(chapter 4.3)
a) Find the intervals of which f is increasing or decreasing
b) Find the local maximum and minimum values of f
c) Find the intervals of concavity
16. An open box is to be made from a rectangular piece of material (2 ft by 3 ft) by
cutting equal squares from each corner and turning up the sides. Find the dimensions of
the box of maximum volume. (chapter 4.7)
17. The top and bottom margins of a poster are each 6 cm and the side margins are each 4
cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions
of the poster with the smallest area. (chapter 4.7)
18. Find the general antiderivative of the function: f(x) = 3cosx – 6sinx (chapter 4.10)
19. Find f given f ( x )  x x , and f(1) = 2
(chapter 4.10)
20. Find a lower estimate and an upper estimate for the area underneath the graph of
f(x) = 4 – x2 from x = 0 to x = 2. Use 4 rectangles. (chapter 5)
21. 12. Evaluate each definite integral. Show your work!!! (chapter 5)
z
5
a.
6 dx
z
2
b.
2
1
3
x4
dx
22. Evaluate each integral. Show all work. (chapter 5)
a.
z
3 xdx
b.
z
(8x 3  3x 2 )dx