Final

Question 1

A limit only exists when

Answer 1

The lim from the right = lim form the left

Question 2

A limit DNE when

Answer 2

Function has a different height from the left and right side of x = c, vertical asymptotes, or if f(c) oscilates between 2 fixed values as x approaces c

Question 3

Process for solving limits

Answer 3

1) Direct Substitution
2) 0/k means lim = 0
3) n/0 means lim DNE
4) 0/0 means do more

Question 4

Methods for 0/0 limits

Answer 4

Factor, Foil, Rationalize (cojugate), getting rid of complex fractions, etc. Always go back to direct sub to finish the problem

Question 5

lim as x approaches 0 of sin(x)/x or sin(kx)/kx

Answer 5

1

Question 6

lim as x approaches 0 of 1 - cosx / x

Answer 6

0

Question 7

lim as x approaches 0 of tanx/x

Answer 7

1

Question 8

Definition of continuity (3 rules)

Answer 8

1) f(c) is defined
2) lim x->c f(x) exists
3) lim x->c f(x) = f(c)

Question 9

Intermediate Value Theorem

Answer 9

If f(x) is continuous on the closed interval [a,b] and k is some height between f(a) and f(b), then there exists some c value between [a,b] such that f(c)=k

Question 10

Slope of the Secant Line

Answer 10

AROC (y2-y1)/(x2-x1)

Question 11

Slope of the Tangent Line (Derivative)

Answer 11

IROC

Question 12

f(x) is not Differentiable When

Answer 12

discontinuous, corner, cusp, or a vertical tangent, different slopes from the left and right

Question 13

f(x) is Differentiable When

Answer 13

1) Continuous
2) Slope from the right = slope from the left

Question 14

Power Rule f(x) = x^n

Answer 14

f’(x) = nx^(n-1)

Question 15

Product Rule d/dx [f(x)g(x)]

Answer 15

f(x)g’(x) + f’(x)g(x)

Question 16

Quotient Rule d/dx [f(x)/g(x)]

Answer 16

[g(x)f’(x)-f(x)g’(x)]/g(x)g(x)
lo di hi - hi di lo over lo lo

Question 17

Chain Rule (Most Important)
d/dx [f(g(x))]

Answer 17

f’(g(x)) g’(x)
Work from the outside and work your way in only one derivative at a time

Question 18

d/dx sinx

Answer 18

cosx

Question 19

d/dx cosx

Answer 19

-sinx

Question 20

d/dx tanx

Answer 20

sec(x)^2

Question 21

d/dx cotx

Answer 21

-csc(x)^2

Question 22

d/dx secx

Answer 22

sec(x)tan(x)

Question 23

d/dx csc

Answer 23

-csc(x)cot(x)

Question 24

Implicit Differentiation

Answer 24

Derive both sides with respect to x. When taking the derivative of a term with y attach a y’ (chain rule). Gather all the y’ on one side and solve for y’

Question 25

Derivatives of Inverse Functions at a point (p,q)

Answer 25

The original function is (q, p) If g(x) is the inverse of f(x) Then g'(p) = 1/f'(q)

Question 26

Equation of the Tangent Line

Answer 26

y = m(x-x1) + y1 where m is the slope of the tangent line (derivative) and (x1, y1) is the point of tangency

Question 27

Tangent Line Approximation

Answer 27

Find the tangent line equation of the given x. Plug in estimate x value into tangent line equation

Question 28

Related Rates

Answer 28

1) Draw a picture 2) Write down the Find and Know 3) Write an equation that matches the variables 4) Differentiate both sides of the equation with respect to time 5) Plug in given information and solve for the needed rate. DO NOT PLUG IN ANYTHING UNTIL AFTER YOU HAVE TAKEN THE DERIVATIVE! 6) Circle the final answer and give the UNITS

Question 29

Volume of a Sphere

Answer 29

4/3 pi r^3

Question 30

Volume of a Cone

Answer 30

1/3 pi r^2 h

Question 31

Volume of a Cylinder

Answer 31

pi r^2 h

Question 32

Critical Numbers

Answer 32

f'(x) = 0 or undefined

Question 33

First Derivative Test

Answer 33

Finding Increasing and Decreasing Intervals Test the intervals between the critical numbers to determine if f'(x) is positive or negative

Question 34

f'(x) > 0 means f(x) is

Answer 34

Increasing

Question 35

f'(x) < 0 means f(x) is

Answer 35

Decreasing

Question 36

Relative Max Occurs When

Answer 36

The derivative of the function changes from positive to negative at a critical number

Question 37

Relative Min Occurs When

Answer 37

The derivative of the function changes from negative to positive at a critical number

Question 38

Absolute Max and Mins (closed interval)

Answer 38

Find f(x) values for endpoints and critical numbers; determine the highest and lowest values

Question 39

Second Derivative Test

Answer 39

Uses concavity to determine if a critical number is a relative max or min

Question 40

f'(c) = 0 and f''(c) > 0 (Positive)

Answer 40

f has a relative min at x = c

Question 41

f'(c) = 0 and f''(c) < 0 (Negative)

Answer 41

f has a relative max at x = c

Question 42

When f''(x)>0

Answer 42

f(x) is concave up

Question 43

When f''(x)<0

Answer 43

f(x) is concave down

Question 44

Point of Inflection

Answer 44

A point where the graph of a function changes concavity f''(x) has to change signs in order for there to be an inflection point

Question 45

Mean Value Theorem

Answer 45

If the function f(x) is continuous on [a,b], AND differentiable on (a,b), then there is at least one number x=c in (a,b) such that f'(c) = f(b) - f(a) / b - a The tangent slope equals the secant slope

Question 46

Intermediate Value Theorem

Answer 46

If a function f(x) is continuous on [a,b] and N is a value between f(a) and f(b), then there exists a value c in (a,b) such that f(c) = N

Question 47

Extreme Value Theorem

Answer 47

If f(x) is continuous on [a,b] then f(x) has both an absolute max and min on [a,b]

Question 48

Fundamental Theorem of Calculus Part 1

Answer 48

If f(x) is continuous on [a,b] and F(x) is an antiderivative of f(x) then integral from a to b of f(x)dx = F(b)-F(a) Top - Bottom

Question 49

Fundamental Theorem of Calculus Part 2

Answer 49

If F(x) = the integral from a to x of f(t)dt then F'(x) = f(x)

Question 50

Rolle's Theorem

Answer 50

If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b) then there exists a c in (a,b) such that f'(c) = 0

Question 51

L'Hopital's Rule

Answer 51

If the lim x->c of f(x)/g(x) results in an indeterminate form and f(x) and g(x) are differentiable then lim x->c of f(x)/g(x) = lim x->c of f'(x)/g'(x)

Question 52

Area of a Square (Cross Section)

Answer 52

b^2

Question 53

Area of a rectangle with h = 1/2b (Cross Section)

Answer 53

1/2 b^2

Question 54

Area of an Isosceles Right Triangle with base as leg (Cross Section)

Answer 54

1/2 b^2

Question 55

Area of an Isosceles Right Triangle with hypotenuse as base ( Cross Section)

Answer 55

1/4 b^2

Question 56

Area of an Equilateral Triangle (Cross Section)

Answer 56

sqrt(3)/4 b^2

Question 57

Area of a Semicircle (Cross Section)

Answer 57

pi/8 b^2