definition of the derivative f’(x) of a function f(x) the...
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Calculus 1501: Practice Exam 1
1. State the following definitions or theorems:
a) Definition of a function f(x) having a limit L
b) Definition of a function f(x) being continuous at x = c
c) Definition of the derivative f’(x) of a function f(x)
d) The “Squeezing Theorem”
e) The “Intermediate Value Theorem”
f) Theorem on the connection of differentiability and continuity
g) Derivatives of sin(x) and cos(x) (with proofs)
2. The picture on the left shows the graph of a certain function. Based on that graph, answer the questions:
a) )(lim1
xfx
b) )(lim1
xfx
c) )(lim1
xfx
d) )(lim0
xfx
e) Is the function continuous at x = -1?
f) Is the function continuous at x = 1?
g) Is the function differentiable at x = -1?
h) Is the function differentiable at x = 1?
i) Is f’(0) positive, negative, or zero?
k) What is f’(-2) ?
3. Find each of the following limits (show your work):
a) 4lim3x
b) 3
2lim
2
3
x
xx
x c)
152
3lim
23
xx
x
x
d) 1
lim1 x
x
x e)
1lim
1 x
x
x f)
1lim
1 x
x
x
g) 2
2
0 3
)(sinlim
x
x
x h)
)(cos
)(sinlim
2
2
0 x
x
x i)
x
x
x 7
)6sin(lim
0
j) )cos(1
lim2
0 t
t
t k) )
1sin(lim
0 xx
x l)
2
2
432
13lim
xx
x
x
m) x
x
x 32
13lim
2
n) xx
x
1lim 2
4. Consider the following function:
0 if ,2
0 if ,)(
2
xx
xxxf
a) Find )(lim0
xfx
b) Find )(lim0
xfx
c) Find )(lim2
xfx
(note that x approaches two, not zero) d) Is the function continuous at x = 0
f) Is
1 if 17
1 if ,1
1)(
2
x
xx
xxf continuous at -1 ? If not, is the discontinuity removable?
g) Is there a value of k that makes the function g continuous at x = 0? If so, what is that value?
0 if )23(
0 if ,2)(
xxk
xxxg
5. Please find out where the following functions are continuous:
a) )2cos()( 2 xxf b) )(sin1
)(2 x
xxf
c)
0 if ,0
0 if ,)(sin
)(
2
x
xx
xxf d)
0 if ,2
0 if ,2
)sin()(
x
xx
xxf
6. Find the value of k, if any, that would make the following function continuous at x = 4.
2 if
2 if 2
4)(
2
xk
xx
xxf
7. Prove that the function 0143 xx has at least one solution in the interval [1, 2]. Also, prove that the
function )cos(xx has at least one solution in the interval ]2/,0[
8. Use the definition of derivative to find the derivative of the function 23)( 2 xxf . Note that we of course
know by our various shortcut rules that the derivative is xxf 6)(' . Do the same for the function
xxf
1
1)( and for xxf )( (use definition!)
9. Consider graph of f(x) you see below, and find the sign of the indicated quantity, if it exists. If it does not
exist, please say so.
f(0)
f’(0)
f(-2)
f’(-2)
f(2)
f’(2)
10. Consider the function whose graph you see below, and find a number x= c such that
a) f is not continuous at x= a
b) f is continuous but not differentiable at x= b
c) f’ is positive at x= c
d) f’ is negative at x= d
e) f’ is zero at x= e
f) f’ does not exist at x= f
10. Please find the derivative for each of the following functions (do not simplify unless you think it is helpful).
xxxxf )sin()( 22
)2()( 42 xxxxf
)1
()( 32
xxxxf
2523)( 35 xxxxf
2
4 32)(
x
xxxf
)sin()( 3 xxxf
)cos()sin()( xxxf
)(sin)( 2 xxf
3
)sin()(
4
x
xxf
4
)sec()(
x
xxf
xxxf )tan()(
6sin)( 2
xf
xx
xxxf
4
32)(
2
4
1)(
2
2
x
xxf
3
)sin()(
x
xxxf
)sin()21(
)cos()(
2
xx
xxxf
)tan()( xxf , find )('' xf
)cos()( xxxf , find )(''' xf
1523)( 35 xxxxf , find )()7( xf
11. Find the equation of the tangent line to the function at the given point:
a) 1)( 2 xxxf , at x = 0
b) xxxf 2)( 3 , at x = 1
12. For the function displayed below, find the following limits:
a) )(lim xfx
b) )(lim xfx
c) )(lim5
xfx
d) )(lim5
xfx
12. Suppose the function 2
4 32)(
x
xxxf
indicates the position of a particle.
a) Find the velocity after 10 seconds
b) Find the acceleration after 10 seconds
c) When is the particle at rest (other than for t = 0)
d) When is the particle moving forward and when backward
14. Find the following limits at infinity:
124
32lim
23
4
xxx
xx
x
1lim
23
5
xxx
xx
x
4
23
32
124lim
xx
xxx
x
3
23
3
1lim
xx
xxx
x
)24)(72(
)1)(43(lim
xx
xx
x
x
x
x
1lim
2
