Math 20A Final Review Outline

Section 4.5: Graph Sketching and Asymptotes

• Know how to use the tables from Sections 4.3 and 4.4 to get a sketch of the graph of f(x)

Sign Combination Curve Type
Increasing and Concave Up
Increasing and Concave Down
Decreasing and Concave Up
Decreasing and Concave Down

• Know what is meant by a transition point

• It is helpful to use the following steps to sketch a curve:

1. Determine the domain of f

2. Determine the signs of f' and f''

3. Note the transition points and sign combinations

4. Draw arcs of the appropriate shapes and asymptotic behavior

• Know what is meant by a horizontal asymptote as well as how to compute it

- Note: A function may have at most two horizontal asymptotes, one corresponding to
each limit

- As an example, find the horizontal asymptotes of the function f(x) = arctan(x)

• Know what is meant by a vertical asymptote as well as how to compute it

• Know how to determine the asymptotic behavior of a rational function

- The asymptotic behavior only depends on the leading terms of the numerator and de-
nominator. If , then

- Evaluate

- Sketch the graph of


Section 4.6: Applied Optimization

• Know the general strategy for solving optimization problems

1. Draw a picture (if not given) of the problem and label variables .

2. Write an equation for the expression that is to be optimized .

3. Write an equation that uses the constraint given.

4. Substitute the constraint into (2) to get an equation with only one variable.

5. Take the derivative of (4).

6. Set the derivative equal to 0 and solve for the variable.

7. Plug the solution from (6) into the constraint equation (3) to get the other variable.

8. Check to see that your answer makes sense.

• Know how to use the above strategy to solve optimization problems

- The difference of two numbers is 18. What is the smallest pos-
sible value for the product of these two numbers?

- You are asked to design a box with a square base and an open
top with a volume of 32,000 cm3. Find the dimensions (base
and height) that minimize the amount of material used.

- A farmer has L feet of fencing with which to fence a rectangular
corral with six compartments (see the figure to the right). What
are the dimensions (x and y) of the corral with maximum area
that the farmer can fence, and what is the maximum area?

• Note: You should also check the endpoints of the interval (provided it is a closed interval) to
see if that provides a max/min. (Typically, they do not.)

• If the interval is open, f does not necessarily have a minimum or maximum value. If it does,
it must occur at a critical point. To determine if a min or a max exists, analyze the behavior
of f as x approaches the endpoints of the interval.

Section 4.7: L'Hopital's Rule

• Know what L'Hopital's Rule says and when we can apply it

If we are considering a limit and we get or , we can instead look at
and these two limits will be equal.

- Calculate

• Note: L'Hopital's Rule still holds even if a = ∞
- Calculate

• Limits involving the indeterminate forms 00, 1, or ∞0 can often be evaluated by first taking
the logarithm and then applying L'Hopital's Rule.

- Calculate

• Note: Sometimes you may need to apply L'Hopital's Rule twice
- Determine

• Know what it means for a function g(x) to dominate another function f(x) as x → ∞ (That
is, In that case, we say that f(x) g(x).

• Know that xn ex for every exponent n

- Show (by applying L'Hopitals' Rule twice) that ex dominates x2 as x → ∞. (That is,
x2 ex.)
- Find the horizontal asymptotes of . Be sure to justify your answer by
evaluating the appropriate limits.

Section 4.8: Newton' s Method
• Know the formula for Newton's Method:

• Given an initial guess, , know how to refine your guess following the above formula. (Try
to choose close to the root.)

- There is a root near 2 of the function f(x) = x3 − 2x − 5. Use Newton's method twice to
find a better approximation for the root.

Section 4.9: Antiderivatives

• Know what is meant by an antiderivative of a function

• Know what the "+C" is at the end of the F(x) and why we can (and need to) put it there

• Know the notation for an indefinite integral

• Know the Power Rule for Integrals

- The best way to remember the formula is to think about undoing the Power Rule for

- As an analogy, consider getting dressed in the morning. You first put on your socks and
then your shoes. In the evening, you take o your shoes and then take o your socks.

- There are two steps to the Power Rule for Derivatives:

1. Multiply by the exponent (this is putting the socks on)

2. Subtract one from the exponent (this is putting the shoes on)

- If we think about "undoing" these steps, we obtain the Power Rule for Integrals

1. Add one to the exponent (this is taking the shoes o )

2. Divide by the exponent (this is taking the socks o )

- Lastly, we add a "+C", since we lost information when we took a derivative

• Note: If you look at the Power Rule, you will notice that something breaks when n = −1.
Remember, the derivative of ln(x) was 1/x, so we have that

• Know why we need to put absolute value signs around x

• Know that Integrals follow the same sum and multiple rules as derivatives

• Know the basic Trigonometric Integrals

For k ≠ 0,
• Know that

• Know what is meant by a differential equation

• Know what is meant by an initial condition and how that affects the antiderivative of a

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