final exam

Question 1

range of ln graphs

Answer 1

(- ∞, ∞)

Question 2

domain of ln functions

Answer 2

(0, ∞)

Question 3

how to find x-intercepts of a graph

Answer 3

sub in 0 for y and solve for x

Question 4

range and domains of inverse functions

Answer 4

range of f = domain of inverse

domain of f = range of inverse

Question 5

how to find the equation of the inverse function given the original function

Answer 5

change x and y and solve for y

Question 6

how to tell if a function has an inverse

Answer 6

it’s a one to one function and passes the horizontal line test

Question 7

what is done differently when evaluating the values of inverse trig functions for cosine and secant?

Answer 7

if cos(a) and a is negative, have to subtract the value from pi

if sec(a) and a is negative, have to add value to pi

Question 8

when testing sign of critical points, what equation do you sub the points back into?

Answer 8

derivative - f’(x)

Question 9

how to find the local minimum and maximum

Answer 9

• find domain first (idk if this is must)
• find critical numbers by finding where the derivative is equal to 0
• create a number line for the sign of the derivative
• positive intervals: f is increasing, negative intervals: f is decreasing
• find the y value by subbing into original f(x) equation to find local extreme values

Question 10

derivative of 3^2t

Answer 10

3^2t times the derivative of the exponent times the ln of base so

3^2t (2) ln(3)

Question 11

slope of tangent and normal lines

Answer 11

slope of the normal line is the perpendicular slope to the tangent line

Question 12

ln (1) =

Answer 12

0

Question 13

when do you reject a critical point value?

Answer 13

when its not in the domain

Question 14

antiderivative of 5^x

Answer 14

5^x ⋅ 1/ln(5)

Question 15

how do you find the limit without shortcuts?

Answer 15

you have to divide the whole term by the highest x power

but if the highest x power is 7x^4, you only divide everything by the x^4 not the 7 too

anything divided by x raised to a power is approaching 0

ex. 6/x^3 approaching - infinity is just 0

Question 16

f is continuous at x = a if what 3 conditions exist?

Answer 16

1. f(a) is defined.
2. lim as x → a f(x) exists.
3. lim x→a f(x) = f(a).

Question 17

how to solve a limit problem with an absolute value

Answer 17

if coming from the right, use the same equation but if coming from the left, have to add a negative to the WHOLE equation in the front

Question 18

how to find antiderivative if u’/u form

Answer 18

if numerator is derivative of the denominator then can just put ln of the absolute value of denominator

Question 19

what does the extreme value theorem state?

Answer 19

a continuous function on a closed interval has to have both an absolute maximum and an absolute minimum on I

Question 20

how to tell if a graph is function

Answer 20

Vertical Line Test

“many to one is okay, one to many is not okay”

Question 21

formula for vertex of parabola

Answer 21

x = -b/2a

(this goes in the middle value of the table when you are finding a graph)

Question 22

can you take the cube root of an odd number?

Answer 22

yes, because you can never get an imaginary number from an odd root!!!!

Question 23

how to find the domain when given a radical?

what ab when radical is in the denominator?

Answer 23

take what’s inside radical and make it greater than or equal to 0 then solve using a number line and plugging in points to check neg. or pos.

• radical in denominator: set equal to greater than 0 bc denominator cannot be 0

Question 24

how to find the y and x intercept and how is each value written

Answer 24

y-intercept: plug in 0 for x and solve for y

x-intercept: plug in 0 for y and solve for x

both written as a coordinate point (x,y) or written as x = (x intercept) or y = (y intercept)

Question 25

Horizontal Line Test

Answer 25

If graph passes horizontal line test, that means it’s a one to one function, and therefore has an inverse function

Question 26

the sine function

Answer 26

goes in increments of 0, π/2, 3π/2, and 2π, up and down to 1 and -1

middle, high, middle, low, middle

Question 27

Cosine graph

Answer 27

goes in increments of 0, π/2, 3π/2, and 2π, up and down to 1 and -1

starts high, middle, low, middle, high

Question 28

finding limit when ln(x) = 0

Answer 28

anytime after subbing in ln(x) = 0, graph is going to - ∞

Question 29

limit when graph oscillates at a point

Answer 29

DNE

Question 30

lim as x approaches infinity of sin(x)

Answer 30

DNE because graph is oscillating

Question 31

version I and version II of limits

Answer 31

version I:
lim x →a f(x) - f(a) / x-a

version II:
lim h→0 f(x+h) - f(x) / h

version II is easier to use especially when you have more complex polynomials

Question 32

corners on f(x) = _______ on f’(x)

vertical tangents on f(x) = ___________ on f’(x)

Answer 32

holes

vertical asymptotes

Question 33

normal line

Answer 33

slope is perpendicular to the slope of the tangent line (so negative reciprocal)

Question 34

how to find say the 93rd derivative of a function

ex. 93rd derivative of cos(3x+4)

Answer 34

write out 3 derivatives because after that cycle just repeats

divide 93 by 4 and the remainder tells you what (derivative) cycle to use (in this case 1 means the first derivative)

• but have to also put that extra 3 from the derivative of the inside raised to the 93rd power too