MATH 121 Formulas- final exam Questions fully solved & verified for accuracy(A+graded)

Definition of derivative

slope of tangent line

Definition of derivative given an x-value and a point

Power rule

f(x)=x^n

f'(x)=nx^n-1

Product rule

Quotient rule

Chain Rule

Quadratic formula

x = -b ± √(b² - 4ac)/2a

When is a function not differentiable?

1. when there is a discontinuity (a jump in the graph)

2. at a corner

3. when it is not defined (at a gap)

First Derivative Test

Determines whether a function is increasing or decreasing

Steps to first derivative test

1. find f'(x)

2. set equal to 0

3. determine x-values, letting the be a, b

4. set up intervals: example: [-infinity, a], [a ,b], [b, infinity]

5. plug in test points within those intervals to f'(x) to determine whether it is decreasing or increasing

f'(x)>0

function is increasing

f'(x)<0

function is decreasing

Second derivative test

Determines whether a function is concave up or down

Steps to second derivative test

1. find double derivative

2. set equal to 0

3. plug in test points to f"(x) to determine concavity

f''(x)>0

concave up

f"(x)<0

concave down

what are points of inflection?

They are points where the graph changes from concave up to concave down, or vice versa

How do you find points of inflection?

They exist where f"(x)= 0 or f"(x) does not exist or when you can use the second derivative test to prove that the grave switches in concavity

How do you find the absolute minimum and maximum of a graph?

1. find critical points

2. plug in critical points and the points from the given interval into f(x) (original equation) to determine y-values

3. absolute min and max are y-values because they are either the highest or lowest points of the graph- the lowest y-value is the minimum, the highest is the maximum

What is he derivative of e^x"?

e^x

vertex formula

-b/2a

Point slope formula

y-y1=m(x-x1)

how do you get the vertical asymptote?

set the bottom to 0 and solve to find what x cannot be: vertical asymptote should be a line in the form of x=a

how do you get the horizontal asymptote

take the limit as x goes to infinity: horizontal asymptote should be line in the form of a y=a

take the derivative of e^2x

e^2x (2)

antiderivative of e^2x

(1/2)e^2x

what is the derivative of f(x)= ln(2x)?

f'(x)= u'/u, or 2/2x

volume by disks

volume by shells

Derivative of ln|x|

1/x

antiderivative of 1/x

ln|x|

formula for exponential growth

Future value= Pe^(rt)

p= Principe amount

r=growth rate

t=time

ln 1

0