Study Guide #1 S2

Question 1

What does it mean if y = f(x) is defined at x = a

Answer 1

f(a) exists

Question 2

An example of a function where y = f(x) is not defined at x = 3

Answer 2

y = 4 / (x - 3)

Question 3

What are the 3 conditions for a function to be continuous on the interval (a, b)

Answer 3

f(a) is defined (that is, a is in the domain of f)

Lim f(x) as x -> a exists

Lim f(x) as x -> a = f(a)

Question 4

Write a function that is continuous on the interval (-2, 4)

Answer 4

y = x

Question 5

What’re the conditions for differentiability

Answer 5

Lim as h -> 0+ = Lim as h -> 0-

Question 6

What does differentiability imply?

Answer 6

Continuity

Question 7

What has to be true for Lim f(x) as x -> a to exist?

Answer 7

Lim f(x) as x -> a+ = Lim f(x) as x -> a-

Question 8

Lim f(x) as x -> a = Infinity

Does the limit exist?

Answer 8

No

Question 9

Lim f(x) as x -> a+ = 4

Does Lim f(x) as x -> a have to exist?

Answer 9

No, because it may disagree at x = a-

Question 10

The 2 definitions of a derivative (used for finding derivatives the slow way)

Answer 10

Lim (f(x+h) - f(x)) / h as h -> 0

Lim (f(x) - f(a)) / a as x -> a

Question 11

Intermediate Value Theorem

Answer 11

Suppose that f is continuous on the closed interval [a, b] and let n be any number between f(a) and f(b) where f(a) can’t equal f(b) then there exists a number c such that f(c) = N

Question 12

Mean Value Theorem

Answer 12

if f is differentiable on the interval [a, b], then there is a number c between a and b such that:
1. f’(c) = (f(b) - f(a)) / b - a
2. f(b) - f(a) = f’(c)(b - a)

Question 13

Squeeze Theorem

Answer 13

if f(x) < f(g) < f(h) when x is near a (except possibly at a) and
Lim f(x) as x -> a = Lim h(x) as x -> a = L
then
Lim g(x) as x -> a = L

Question 14

Interval Method

Answer 14

to find the absolute max and min values on a continuous function f on a closed interval [a, b]
1. find the values of f that are critical numbers on [a, b]
2. find the values of f at the endpoints of the interval
3. the largest value from steps 1 and 2 is the absolute max and the smallest is the absolute min.

Question 15

Derivative of x^n

Answer 15

nx^n-1

Question 16

Derivative of fg

Answer 16

f’g + fg’

Question 17

Derivative of f/g

Answer 17

f’g - fg’ / g^2

Question 18

Derivative of f(g(x))

Answer 18

f’(g(x))g’(x)

(Chain rule)

Question 19

Derivative of sin(x)

Answer 19

cos(x)

Question 20

Derivative of cos(x)

Answer 20

-sin(x)

Question 21

Derivative of tan(x)

Answer 21

sec^2(x)

Question 22

Derivative of cot(x)

Answer 22

-csc^2(x)

Question 23

Derivative of sec(x)

Answer 23

sec(x)tan(x)

Question 24

Derivative of csc(x)

Answer 24

-csc(x)cot(x)

Question 25

Derivative of e^x

Answer 25

e^x

Question 26

Derivative of a^x

Answer 26

a^xln(a)

Question 27

Derivative of ln(x)

Answer 27

1/x

Question 28

Anti-Derivative of x^n

Answer 28

(x^n+1 / n+1) + c

Question 29

Anti-Derivative of 1/x

Answer 29

ln|x| + c

Question 30

Anti-Derivative of a^x

Answer 30

(a^x / ln(a)) + c

Question 31

Anti-Derivative of ln(x)

Answer 31

xln(x-x) + c

Question 32

Anti-Derivative of tan(x)

Answer 32

ln|sec(x)| + c

Question 33

Anti-Derivative of cot(x)

Answer 33

ln|sec(x) + tan(x)| + c

Question 34

Anti-Derivative of csc(x)

Answer 34

ln|csc(x) - cot(x)| + c

Question 35

Anti-Derivative of sec^2(x)

Answer 35

tan(x) + c

Question 36

Anti-Derivative of sec(x)tan(x)

Answer 36

sec(x) + c

Question 37

Anti-Derivative of csc^2(x)

Answer 37

-cot(x) + c

Question 38

Anti-Derivative of csc(x)cot(x)

Answer 38

-csc(x) + c

Question 39

Anti-Derivative of tan^2(x)

Answer 39

tan (x) - x + c