Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 10
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41Worksheet # 8: Review for Exam I
1. Find all real numbers of the constant a and b for which the function f (x) = ax + b satisfies:
(a) f
f (x) = f (x) for all x.
(b) f
f (x) = x for all x.
2. Simplify the following expressions.
(a) log
125
5
(b) (log
16)(log
2)
4
4
(c) log
75 + log
3
15
15
x
(d) log
(x(log
y
))
x
y
(e) log
(1
cos x) + log
(1 + cos x)
2 log
sin x
π
π
π
3. Suppose that tan(x) = 3/4 and
π < x < 0. Find cos(x), sin(x), and sin(2x).
2x+5
4. (a) Solve the equation 3
= 4 for x. Show each step in the computation.
1
3
(b) Express the quantity log
(x
2) +
log
(x)
log
(5x) as a single logarithm. For which x is the
2
2
2
3
resulting identity valid?
5. Calculate the following limits using the limit laws. Carefully show your work and use only one limit
law per step.
(a) lim
(2x
1)
x
0
x + 4
2
(b) lim
x
x
0
6. Determine if the following limits exist and calculate the limit when possible.
x
2
2
x
(a) lim
(c) lim
1
1
x
1
x
2
x
2
x
2
x
2
(d) lim
2
x
16
x
4
x
2
x + 1
(b) lim
(e) lim
1
1
x
2
x
2
x
2
x
2
2
7. Suppose that the height of an object at time t is h(t) = 5t
+ 40t.
(a) Find the average velocity of the object on the interval [3, 3.1].
(b) Find the average velocity of the object on the interval [a, a + h].
(c) Find the instantaneous velocity of the object time a.
sin(x)
x
8. Use the fact that lim
= 1 to find lim
.
x
sin(3x)
x
0
x
0
9. (a) State the Squeeze Theorem.
1
(b) Use the Squeeze Theorem to find the limit lim
x sin
2
x
x
0
10. Use the Squeeze Theorem to find lim
cos x cos(tan x)
x
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