final exam Flashcards
range of ln graphs
(- ∞, ∞)
domain of ln functions
(0, ∞)
how to find x-intercepts of a graph
sub in 0 for y and solve for x
range and domains of inverse functions
range of f = domain of inverse
domain of f = range of inverse
how to find the equation of the inverse function given the original function
change x and y and solve for y
how to tell if a function has an inverse
it’s a one to one function and passes the horizontal line test
what is done differently when evaluating the values of inverse trig functions for cosine and secant?
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
when testing sign of critical points, what equation do you sub the points back into?
derivative - f’(x)
how to find the local minimum and maximum
- 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
derivative of 3^2t
3^2t times the derivative of the exponent times the ln of base so
3^2t (2) ln(3)
slope of tangent and normal lines
slope of the normal line is the perpendicular slope to the tangent line
ln (1) =
0
when do you reject a critical point value?
when its not in the domain
antiderivative of 5^x
5^x ⋅ 1/ln(5)
how do you find the limit without shortcuts?
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
f is continuous at x = a if what 3 conditions exist?
- f(a) is defined.
- lim as x → a f(x) exists.
- lim x→a f(x) = f(a).
how to solve a limit problem with an absolute value
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
how to find antiderivative if u’/u form
if numerator is derivative of the denominator then can just put ln of the absolute value of denominator
what does the extreme value theorem state?
a continuous function on a closed interval has to have both an absolute maximum and an absolute minimum on I
how to tell if a graph is function
Vertical Line Test
“many to one is okay, one to many is not okay”
formula for vertex of parabola
x = -b/2a
(this goes in the middle value of the table when you are finding a graph)
can you take the cube root of an odd number?
yes, because you can never get an imaginary number from an odd root!!!!
how to find the domain when given a radical?
what ab when radical is in the denominator?
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
how to find the y and x intercept and how is each value written
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)
Horizontal Line Test
If graph passes horizontal line test, that means it’s a one to one function, and therefore has an inverse function
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
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
finding limit when ln(x) = 0
anytime after subbing in ln(x) = 0, graph is going to - ∞
limit when graph oscillates at a point
DNE
lim as x approaches infinity of sin(x)
DNE because graph is oscillating
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
corners on f(x) = _______ on f’(x)
vertical tangents on f(x) = ___________ on f’(x)
holes
vertical asymptotes
normal line
slope is perpendicular to the slope of the tangent line (so negative reciprocal)
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
