Summary MATH 100 midterm exam review

MATH 100 midterm review problems:

1. Find the domain of f if
 
p 2 5 3
f (x) = 3 − |x − 3| + ln −x − x +
8 8

2. Find the domain of
 
3
f (x) = ln − |x − 1| + |x − 2| − |x − 3|
2

3. Evaluate the following limits or explain why it does not exist:

tan2 (x + 1)
i. lim
x→−1 (x2 − 1) sin(4x + 4)
 2 
x − 2x − 3
ii. lim− arctan
x→1 x−1
sin2 (x)
iii. lim p
x→0 1 + x sin(x) − cos(x)
|x − 1|
iv. lim 2
x→1 x − 1

√
4. What is f −1 (0) if f (x) = 12 ln(x + 1) − ln x + 2.


5. Use
√ the definition of the derivative as a limit to find the derivative of
f (x) = x
3




6. Use Mathematical induction to show that:

1 + 3 + 5 + ... + (2n − 1) = n2
for all positive integers n .

7. Use induction to show that:
1 1 1 1 1
+ 2 + 3 + ... + n = 1 − n
2 2 2 2 2
for every positive integer n.

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