150-sampleexam1s07
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Comments re sample exams: no sample ever contains every example of every problem type possible. Use this to get an idea of the length and level of the exam. Use your study guide to besure you are fully prepared!
Sample Exam 1, Calculus I, Spring 07
Directions: Show, in an organized fashion, any work you want considered. Erase or cross outscratch work that you do not want considered. (Partial credit will be awarded for partially
correct work or reasoning.)
12 1a) For the function f whose graph is given, state the value or that it does not exist (DNE).
)(lim2
x f x −−→
)(lim2
x f x +−→
)(lim2
x f x −→
f (-2)
)(lim0
x f x −→
)(lim0
x f x +→
)(lim0
x f x→
f (0)
)(lim1
x f x
−→)(lim
1 x f
x+→
)(lim1
x f x→
f (1)
8 b) Analyze the continuity (or discontinuity) of the graph at x = -2, x = 0, and x = 1. Explain interms of limits, that is, by making clear reference to the definition of continuity.
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Calculus I, Drs. Nogin and Wyels, Fall 2006, Spring 20074
9 6. Use the values given in the table to find the derivatives indicated. Show intermediate steps. x 0 1 2 3 4
)( x f 4 1 1 -3 4)( x g 3 -2 2 -2 3
)( x f ′ 0 -1 -3 -1 2
)( x g ′ -2 1 4 0 1
a) )()()( x g x f x H = ; find )2( H ′
b))(
)()(
x g
x f x J = ; find )2( J ′
c) ( ))()( x g f x K = ; find )0( K ′
12 7. Find derivatives of the following functions. Show your work or briefly (2 – 10 words) giveyour reasoning.
a) t t y =
b)2
6)(
2
+
−−=
x
x x x f
c) 20cos)( 3+−= x x x g π
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Calculus I, Drs. Nogin and Wyels, Fall 2006, Spring 20075
10 8. An astronaut standing on the moon throws a ball straight up into the air with an initialvelocity of 40 m/s. The height of the ball in meters after t seconds is given by
2240)( t t t s −= .
a) Find the average velocity of the ball between 1 second and 3 seconds.
b) Find the instantaneous velocity of the ball at 3 seconds.
10 9. Arrange the following numbers in increasing order and briefly explain your reasoning:
0, )1( f ′ ,2
)1()3( f f −, )1()2( f f −
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Calculus I, Drs. Nogin and Wyels, Fall 2006, Spring 20076
10 10. Show that there is a root of the equation 1)( 3−+= x x x f in the interval (0,1). Show
work/ give reasoning.
10 11. Find the value of c that makes the function continuous on the interval ( )∞∞− , . Show
work/ give reasoning.
∞∈+−
−∞∈+=
),2()3(
]2,(3)( 2
xc x
xcx x f
5 Extra Credit: The graph shown is that of the function2
1)(
x x f = . Note that 25.0)(lim
2=
→
x f x
.
Find a number δ such that 1.025.0)( <− x f whenever δ <−2 x . Sketch or write
enough to indicate your reasoning.