Some Frequently Asked Questions in AP Calculus AB

Note: the ones that don’t look like hyperlinks are not yet answered. They may never be, at this point, but what’s here might be useful to you.

Precalculus (and earlier) concepts

Limits

Differentiation

Tangent Lines

Kinematics (position, velocity, and acceleration)

Analyzing Graphs using Derivatives

Errors

Technology

Related Rates

I might eventually add to this in the future. In particular, right now it only includes material from the first half of the course. If you would like to suggest a question for this FAQ, write me here. I cannot promise to include your question, though.

Precalculus (and earlier) concepts

What does an equation of a line look like?

Well, there are a lot of possiblities. However, the most useful form in this course is called point-slope form. True to its name, it requires two distinct facts: the coordinates of a point located on the line, and the slope of the line. It looks like this: y – y₁= m(x – x₁). The value of m is the slope, and (x₁, y₁) are the coordinates of the point. If you’re asked to find an equation of a line in calculus, this will usually be the simplest way to go about it.

One of the peculiar mistakes calculus students sometimes make that never happens in earlier courses is the use of the formula for the slope (a/k/a the derivative) in the place where a numerical value of slope is supposed to go. Even worse, every once in a while that’s actually a useful thing to do! However, you can do a quick mental check of your answer to avoid this issue. Is the answer supposed to be a line? Then it had better not have any powers on the variable!

How do radicals and fractional exponents work?

How do negative exponents work?

How does interval notation work?

The symbols involved are the left and right parenthesis marks and the left and right square brackets. Parentheses are used to denote < and >, while brackets are for ≤ and ≥. Here are a few examples.

Note that the endpoints of the interval are always written in increasing order. You don’t get a choice here, the smaller value always comes first.

Limits

What type of answer are you looking for when you “evaluate a limit”?

The word “evaluate” literally means to get a number from something. (E = “from” or “out of,” like in “E pluribus unum.”) So to evaluate a limit means to find a numerical result. It’s the y-value that the function approaches (maybe even reaches) as x gets increasingly close to the value it is approaching.

Differentiation

What does “differentiable” mean?

A function is differentiable when it’s derivative exists (“differentiable” = able to be differentiated). In particular, at a place where a function is differentiable, “zooming in” really close will look like a line; the line that you see is basically the tangent line, and its slope is the slope of the function, also called the value of the derivative at that point. Almost all of the functions we work with are differentiable almost all of the time.

What’s probably more useful is knowing what it looks like when a function is not differentiable. There are four ways a function can fail to have a derivative at a point. The most common one is that the function could be discontinuous at that x-coordinate. If the graph is “disconnected,” you certainly won’t see a line when you look very closely. The other ways are if there is a corner, a cusp, or a vertical tangent at the location. The reason corners and cusps make differentiability collapse is that the slope from the left and the slope from the right are different from each other. Vertical tangents, on the other hand, do look like lines when you look ever closer; they look like vertical lines. But since the slope of a vertical line is undefined, the value of the derivative is undefined there, too.

When do you need the product rule?

The product rule is necessary whenever the function you want the derivative of is made of two other functions multiplied together. You don’t need it when one of the two factors is just a number. For instance, the derivative of y = x sin x requires the product rule since both x and sin x are functions with their own (non-zero) derivatives. However, y = 3 sin x does not, since 3 is just a coefficient. Its derivative will be 3 times the derivative of sine. Note that π and e qualify as just-numbers, while θ is a variable, so y = θ sin θ is just like y = x sin x, and needs the product rule.

How does the chain rule work?

The chain rule says that to take the derivative of a composite function in the form f(g(x)), you need the derivative of f (still with g inside) times the derivative of g itself. In symbols, this looks like  There are a couple of important points here. First of all, the derivative of the inside function, g´, is multiplied at the end; it does not go where g was originally. In fact, that brings up the other important point: the original inside function (that’s still g) stays where it was, right inside the first function.

I like to think of this formula as “f-prime of stuff times the derivative of the stuff.” And “stuff” is g(x). But that’s just me. Lots of people say that it’s “the derivative of the outside times the derivative of the inside.”

When do you have to use the chain rule?

Whenever there’s an inside function. (See How does the chain rule work? for what an inside function is.) Now, of course, the problem is how to recognize the existence of an inside function. The most common and easiest to locate versions have parentheses, with their little arms hugging the inside function with all their might. If what’s inside the parentheses is anything other than just a single variable, then you’re probably looking at a situation calling for the use of the chain rule.

The problem says to find a rate of change. What’s that?

It’s a derivative.

L’Hôpital’s rule versus the quotient rule — what’s the difference?

The quotient rule is used to find the derivative of an expression that’s a fraction — a quotient — of two other functions. It’s the one we use the lo-d-hi mnemonic for. L’Hôpital’s rule, on the other hand, is used to find limits of expressions that are the quotients of two other functions. So the functions might look exactly the same, and that’s why the confusion arises. In L’Hôpital’s rule, you don’t use the lo-d-hi formula; you simply take the derivative of the numerator and the derivative of the denominator separately, then again try substitution to evaluate. It’s a lot less work, actually. L’Hôpital’s rule is only used when the original fraction evaluates to either or

Sometimes you write “r. o. c” in the notes. What’s up with that?

“R. O. C.” (capital or lowercase) is a shorthand for “rate of change.” Writing it gets old. In a related abbreviation, “w/r/t” and “wrt” mean “with respect to,” as in “velocity is the r. o. c. of position w/r/t time.”

Tangent Lines

How do you get an equation for a tangent line?

To find an equation of any line, you generally want a point and a slope. The point is determined by the original function, and the slope comes from the derivative.

Example: Find an equation of the tangent line to f(x) = x³ + 2x + 1 at the point where x = 2.

Solution: First, find the point. The x-coordinate is given as 2. To find the y-value, substitute into the function:
f(2) = 2³ + 2(2) + 1 = 8 + 4 + 1 = 13. So our point is (2, 13).

Second, find the slope. The slope is determined by the derivative of the function: f ′(x) = 3x² + 2. The formula itself is not the slope; you still have to get a number. So use the given x-coordinate: f ′(2) = 3(2)² + 2 = 3(4) + 2 = 14, the slope of the tangent line.

Finally, write the equation. In general, the easiest way to do this is to use the point-slope form of a linear equation:
y – y₁ = m(x – x₁). Substituting (2, 13) for the point and 14 for the slope gives the equation we need: y – 13 = 14(x – 2).

How can it be harder than this? Well, the function I chose for the example was pretty simple, and has an easy derivative; not all functions are so easy to differentiate. Sometimes rather than being told a point or an x-coordinate, you might be asked to find the tangent that has a particular slope, or one that passes through some given point not on the function.

What does it mean to have a horizontal tangent line? How do you find a horizontal tangent line?

A horizontal tangent line is a tangent line with a slope of 0. If you are asked to find the equations of any horizontal tangent lines to a function, you’re going to need to know the derivative, since that’s the formula for the slope. Set that formula equal to zero and solve to find the x-coordinate of the point. Use your newly-found x-coordinate to find the y-coordinate in the original function, and then write down the equation of the horizontal line through that y-coordinate.

Example: Find the equations of any horizontal tangent lines to the graph of y = x³ – x² – 5x.

Solution: Horizontal means a slope of zero, and the slope of a function is found using its derivative. So first, we find the derivative:
f ′(x) = 3x² – 2x – 5.

To determine where this function has a zero slope, solve the equation f ′(x) = 0:
3x² – 2x – 5 = 0
(3x – 5)(x + 1) = 0
3x – 5 = 0 or x + 1 = 0
x =  or x = –1

So those are the x-coordinates of the points where there are horizontal tangent lines. They are not, however, the tangent lines themselves. Horizontal lines have equations that look like y = k, where k is a constant. So to get the tangent lines, we need the y-coordinates of the points. Use the formula for the function that was given initially.

When

When x = –1, y = (–1)³ – (–1)² – 5(–1) = 3.

And those are the desired tangent lines: y =  and y = 3.

What does it mean to have a vertical tangent line?

A vertical line is said to have “no slope.” In calculating slope as change in y divided by change in x, a vertical line will have a non-zero change in y with a zero change in x. In other words, there’s a rise, but no run. Since the slope of a tangent line is determined by the derivative, this works out to mean that the value of the derivative has a non-zero numerator with a zero denominator.

Kinematics (position, velocity, and acceleration)

How do you calculate velocity?

Velocity is the rate of change of position. In other words, the velocity function is the derivative of the position function. So if you’re asked to find the velocity of a particle at time t = 3, take the derivative of the position function and substitute in 3 for t. On the other hand, sometimes you are asked to find the velocity at any time t. In that case, you’re just looking for the formula. Take the derivative of position, and leave that formula as your answer.

What’s the difference between instantaneous velocity and average velocity?

Instantaneous velocity is the velocity right now, like what the speedometer in a car registers. Average velocity is the “rate” in the formula distance = rate · time; it’s what you mean when you say that to get 150 miles in 3 hours, you’ll have to drive 50 miles per hour. On that three-hour trip, you’re not very likely to be traveling exactly 50 miles per hour the whole time. You’ll be stopped at the beginning and end, for instance, and there will be speeding up and slowing down quite often along the way, perhaps with a few intermediate stops for red lights and McDonald’s breaks.

To calculate average velocity, just divide change in position by change in time. To calculate instantaneous velocity, take the derivative of the position function and substitute the appropriate value of t.

How do you find acceleration?

Acceleration is the rate of change of velocity. That makes it the second derivative of the position function. If you already have a velocity function, take its derivative to get a formula for finding acceleration. If you only have the position function, you’ll have to take the derivative twice to get to acceleration. (Note that the difference between instantaneous acceleration and average acceleration works like the same thing with velocity. The instantaneous version is a derivative, and the average version would be change in velocity divided by change in time.)

What’s the difference between displacement and distance?
What does the position function tell you?

Analyzing Graphs using Derivatives

How do you know if a quantity is increasing or decreasing? (rate of change)

What does g‘ tell you about g?

How do you find the maximum (or minimum) value of a function?

What are critical numbers? How do you find critical numbers?

How do you make a number line to test for increasing and decreasing? (including labeling)

How much do I have to write down when I find critical numbers?

How much do I have to write down when I find where a function is increasing or decreasing?

How much do I have to write down when I find extreme values of a function?

How do you figure out when the derivative (or second derivative) is positive (or negative)?

How do I find out where a function is concave down or concave up?

Errors

What kinds of errors should I be especially careful to avoid?

Technology

When can I use the dy/dx or nDeriv command on my calculator instead of working out the formula for the derivative?
How do you solve an equation using your calculator?

Related Rates

What formulas do I need to know for related rates problems?

For sure, you have to know that the area of a circle is  A = πr² (and not 2πr). Here is a list. Note that the AP Calculus exam does not provide a “formula sheet” for reference. Often, formulas for cones and spheres have been provided in the problems, but I cannot guarantee that in the future. Don’t panic at the length of the list; you know almost all of them already, but it’s a good idea to read through them every once in a while.