final exam Flashcards

1
Q

range of ln graphs

A

(- ∞, ∞)

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2
Q

domain of ln functions

A

(0, ∞)

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3
Q

how to find x-intercepts of a graph

A

sub in 0 for y and solve for x

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4
Q

range and domains of inverse functions

A

range of f = domain of inverse

domain of f = range of inverse

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5
Q

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

A

change x and y and solve for y

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6
Q

how to tell if a function has an inverse

A

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

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7
Q

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

A

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

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8
Q

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

A

derivative - f’(x)

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9
Q

how to find the local minimum and maximum

A
  • 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
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10
Q

derivative of 3^2t

A

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

3^2t (2) ln(3)

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11
Q

slope of tangent and normal lines

A

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

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12
Q

ln (1) =

A

0

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13
Q

when do you reject a critical point value?

A

when its not in the domain

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14
Q

antiderivative of 5^x

A

5^x ⋅ 1/ln(5)

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15
Q

how do you find the limit without shortcuts?

A

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

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16
Q

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

A
  1. f(a) is defined.
  2. lim as x → a f(x) exists.
  3. lim x→a f(x) = f(a).
17
Q

how to solve a limit problem with an absolute value

A

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

18
Q

how to find antiderivative if u’/u form

A

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

19
Q

what does the extreme value theorem state?

A

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

20
Q

how to tell if a graph is function

A

Vertical Line Test

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

21
Q

formula for vertex of parabola

A

x = -b/2a

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

22
Q

can you take the cube root of an odd number?

A

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

23
Q

how to find the domain when given a radical?

what ab when radical is in the denominator?

A

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
24
Q

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

A

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)

25
Horizontal Line Test
If graph passes horizontal line test, that means it’s a one to one function, and therefore has an inverse function
26
the sine function
goes in increments of 0, π/2, 3π/2, and 2π, up and down to 1 and -1 middle, high, middle, low, middle
27
Cosine graph
goes in increments of 0, π/2, 3π/2, and 2π, up and down to 1 and -1 starts high, middle, low, middle, high
28
finding limit when ln(x) = 0
anytime after subbing in ln(x) = 0, graph is going to - ∞
29
limit when graph oscillates at a point
DNE
30
lim as x approaches infinity of sin(x)
DNE because graph is oscillating
31
version I and version II of limits
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*
32
corners on f(x) = _______ on f'(x) vertical tangents on f(x) = ___________ on f'(x)
holes vertical asymptotes
33
normal line
slope is perpendicular to the slope of the tangent line (so negative reciprocal)
34
how to find say the 93rd derivative of a function ex. 93rd derivative of cos(3x+4)
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
35