Final Exam

Question 1

Rule for a limit to exist

Answer 1

Must come together at the same y value

Question 2

How do you evaluate a limit algebraically

Answer 2

Substitute the x -> number into x
*may need to switch negative sign to positive

Question 3

limit 0/0

Answer 3

indeterminate

Question 4

limit nonzero/0

Answer 4

find vertical asymptote

Question 5

What does x-9 / √x-3 equal?

Answer 5

(√x+3)(√x-3)/(√x-3)

Question 6

If the degree of the denominator of a function is higher then …

Answer 6

The limit equals 0

Question 7

How do you write a limit given a graph

Answer 7

1) Write lim f(x)
2) The “x approaches” part approaches the x intercept on the axis
3) It will equal the y-intercept of the x-intercept on the line

Question 8

A function is continuous if…

Answer 8

If f(c) is defined
lim f(x) exists
x–> c
lim f(x) = f(c)
x –> c

Question 9

How do you solve “slope of a line tangent to the graph for a specified value” problems?

Answer 9

Find the derivative, and solve for x, which equals the slope
The (x,y) is the x = number, and the y = input x into the original function

Question 10

How do you solve chain rule problems given y and u?

Answer 10

1) Get the derivative of y and u
2) Multiply them together
3) Substitute u back into the equation

Question 11

How can you find a derivative using the definition of the derivative?

Answer 11

[f(x+h) - f(x)] / h
Substitute the function into x and solve for x = number

Question 12

What does it mean by how much something will change during the sixth year?

Answer 12

C(6) - C(5)

Question 13

Product rule

Answer 13

[f(x)g(x)]’ = f’(x)g(x) + f(x)g’(x)

Question 14

Quotient rule

Answer 14

[f(x)/g(x)] = [f’(x)g(x) - f(x)g’(x)] / [g(x)] squared

Question 15

How do you differentiate functions with multiple pieces?

Answer 15

Distribute using FOIL if possible

Question 16

What should you consider if you are stuck on log/exp derivatives?

Answer 16

Using the product or chain rule

Question 17

How do you do implicit differentiation?

Answer 17

1.) Take derivative and “fly the flag” with y’ wherever y occurs
2.) Get y’ on one side

Question 18

What should be done given the related rate problem?

Answer 18

1.) Take the derivative using Δx and Δy to “fly the flag”
2.) Get Δy on one side
3.) Plug in values
*If you are missing a value you may need to plug price or quantity into the equation

Question 19

How do you find critical numbers?

Answer 19

Set first derivative = 0

Question 20

How can you find where the function is increasing/decreasing?

Answer 20

Use sign diagram on the critical values

Question 21

How can you find the points where max/mins are?

Answer 21

( critical point f’, f(critical point f’))

Question 22

How can you determine where a function is concave up/down?

Answer 22

1.) Set second derivative = 0
2.) Use sign diagram to find where f’’ is concave up/down

Question 23

How do you get inflection points?

Answer 23

(critical value f’’, f(critical value f’’))

Question 24

Second derivative test

Answer 24

1.) Get zeros of f(x)
2.) Plug into second derivative

If f’‘(c) < 0 concave down, max
If f’‘(c) > 0 concave up, min

Question 25

How do you get Horizontal Asymptote and Vertical Asymptote?

Answer 25

HA: ratio of leading coefficients (if degree is same) VA: set denom = 0

Question 26

How do you get the x and y intercepts?

Answer 26

x int: numerator = 0 y int: set x = 0

Question 27

How do you do practical optimization given a rectangle problem?

Answer 27

Use perimeter and area formulas with information Substitute so there is one variable Find the derivative and get the CV Plug the CV into the second derivative to find if it is a min or max

Question 28

If f(x) = e to the x, then f’(x) =

Answer 28

e to the x

Question 29

If f(x) = e to the f(x), then f’(x) =

Answer 29

e to the f(x) times f’(x)

Question 30

If f(x) = lnx, then f’(x) =

Answer 30

1/x

Question 31

If f(x) = lnf(x), then f’(x) =

Answer 31

1/f(x) times f’(x)

Question 32

When solving for x, what should you keep in mind?

Answer 32

Multiply it by whatever value it represents

Question 33

How do you solve partial derivatives?

Answer 33

1) Find fx and fy 2) For fx, take the derivative of the x pieces and let y pieces equal 0, but if there are y pieces attached to an x, leave them as is 3) Do the opposite for y

Question 34

How do you get marginal cost and revenue?

Answer 34

Take the respective derivative

Question 35

How do you get R(x) given p(x) [price]?

Answer 35

x times p(x) = R(x)

Question 36

How do you solve for marginal vs actual costs given n?

Answer 36

For marginal, take the derivative of (n-1) For actual its C(n) - C(n-1)

Question 37

How do you find absolute extrema in optimization?

Answer 37

1) Take the derivative 2) Find critical values 3) Plug those into the equation along with the given greater than/equal than to find the points

Question 38

How do know what is a max or min in absolute extrema

Answer 38

Lowest y value - abs min, highest y value - abs max

Question 39

What number is exceeded by its square root by the largest amount?

Answer 39

x-{square root of x) - get derivative, CV, solve for max with second derivative Should be CV at 1/4

Question 40

Why will the perimeter of a square always be larger than that of a rectangle?

Answer 40

p = 2(x+y) and y = (1/2)(p-2x) A = xy = x[(1/2)(p-2x)] = -x squared + (1/2)px A' = -2x +(1/2)p = 0 when x = p/4 Sinc A'' = -2 > 0 the max occurs when x= p/4, that is when the rectangle is a square

Question 41

∫nx to the a

Answer 41

[nx to the (a+1)]/[a+1] + C

Question 42

∫e to ax

Answer 42

ae to the x + C

Question 43

∫ (a/x)

Answer 43

a ln of the absolute value of x

Question 44

Two things to do in U Substitution

Answer 44

1. Find u, which is the most complicated part of the problem 2. Find du, which is the derivative of u

Question 45

Do not forget…

Answer 45

To add + C at the end of the solved integral

Question 46

How do you solve definite integrals?

Answer 46

Solve for the integral Plug the top value into the integral minus the bottom value plugged into the integral

Question 47

How do you get the bounds of a definite integral given two functions?

Answer 47

Set them equal to each other and solve for x

Question 48

Willingness to Spend/Consumer Surplus Equation

Answer 48

q0 l D(q)dq 0

Question 49

Consumer Surplus Equation

Answer 49

q0 l D(q)q - p0q0 0

Question 50

Lim e to the x = x – ∞

Answer 50

+∞

Question 51

Lim e to the -x = x – ∞

Answer 51

0

Question 52

Lim (x to the P)/e to the x x – ∞ P = any power

Answer 52

0

Question 53

Lim ln(x) x – ∞

Answer 53

+∞

Question 54

How do you solve equations with infinity as the upper bound?

Answer 54

1) replace ∞ with N, and as N approached ∞ 2) Evaluate integral 3) Try to solve it so that N = 0, or the useful limits