Are you a student thinking about taking an AP Calculus class sometime in the future but unsure whether it’s a good choice for you? If this is the quandary you find yourself in, take a moment to ask yourself these three important questions.
Should I Take AP Calculus?
1. Do I have a passion or willingness to succeed in math?
Although people who do not have a liking for math are still able to do well in this class, people who really enjoy math usually have an easier time. They not only find the lessons and work more pleasurable — maybe even exciting — but also generally have a greater willpower to reach success.
However, keep in mind that everyone’s definition of success is different, and it is important to ask yourself what you wish to get out of this experience. If your sole definition of success is to “get a 5 on the AP exam” but the idea of doing more math brings you frustration or even terror, this is probably not the best course for you. Similarly, if your only goal when signing up for such a course is to be in a class with your friends, you may want to take a step back and find another incentive to take the class. Signing up for a high level math course just because you wish to boost your college application or because you want to have fun with all your peers can be a risky move, maybe even counterproductive, and should be given further consideration.
Remember, there are plenty of other ways to achieve either of those two goals without taking this course, so be sure to make a decision that will make learning manageable and enjoyable, just as it should be.
2. Am I ready for the rigor of this class?
This is a crucial question to ask yourself as AP Calculus is no easy course. In general, AP courses are not easy as they give high-schoolers a taste of college-level material, and AP Calculus is certainly no exception. Some people have been implanted with the idea that AP Calculus is a relatively easy class seeing that its AP exam passing rates are typically higher than most other courses. However, these statistics may not be the best indicators for the difficulty of a particular AP class; rather it is important to look at the content covered in the course. In AP Calculus, you are expected to utilize what you have learned in the context of real-world situations and provide justifications for some of your answers. There is a bigger emphasis placed on applications of skills instead of strict memorization, which may or may not be something you are used to. Additionally the concepts taught in this course will most likely be a little more difficult to grasp than those in math classes you have taken before. If any of this sounds daunting to you, talk to your math teacher, counselor, or someone who has taken the class before to see if it is right for you.
3. Do I have the sufficient background knowledge?
Since calculus in high school builds off of knowledge and skills attained in previous classes like algebra, geometry, and precalculus, it is important to have a somewhat solid understanding of the key concepts taught in those classes.
Next we will quickly review some of the most important topics and skills from prerequisite classes that you should feel comfortable with. If any of it sounds unfamiliar or feels hazy, be sure to have it down before taking calculus! Note that this is not a comprehensive review but rather a memory-jogger of a few important things you should already know.
One of the important algebra skills you will need in AP Calculus is factoring, the splitting up of an expression into the product of multiple factors. This is pretty easy if you are just working with a number or quantity; however, it’s a little more tricky when it comes to expressions with variables or polynomials. There are several methods that you can take when factoring polynomials.
Taking out the GCF or Greatest Common Factor
In order to do this, first find a factor that is present in all the terms. You can then “take out” this factor from the expression by dividing it out from all the terms and then writing your expression as a product. You can continue this process until there are no more common factors among the terms. Here is an example:
Take out the GCF from 4x2+8x.
Since there is a common factor of 4 among all the terms, we can factor it out and rewrite the expression as the product
We notice that there is another common factor among the terms inside the parentheses, namely x, so we take that out as well and write the expression as
Seeing that there are no more common factors among the terms, we are finished.
This is an important skill to have, especially since many higher degree polynomials you will see on tests can be broken up into a product of a quadratic and some other expression, and it will be extremely useful to know how to further break down these quadratics! Remember that quadratics are polynomials with a degree of 2; in other words, the leading term, or the term with the highest power of x, has an exponent of 2.
Many methods have been introduced for factoring quadratics such as the box method, the diamond method, or the slip-slide method, and I would advise you to check these all out before settling on one. There is no “right” method, but make sure you choose the one you are most comfortable with.
Sometimes, however, it is easier to go with “educated guessing” as your strategy for factoring quadratics, especially since tests (like the AP Calculus exam) generally give easier polynomials to factor. Below is an example.
Factor the quadratic 2x2+11x+5.
Let’s first look at what a factored quadratic would look like. We know that a 2nd degree polynomial can be broken up into factors of degree 1, or linear factors.
So we write our factored expression in the form (ax+b)(cx+d) where a, b, c, and d are constants.
When we expand this using distribution or the FOIL method and group like terms, we end up with the expression
(ac) x2+ (bc+ad) x + bd.
Looking back to the quadratic we would like to factor, we notice that
ac = 2, bc+ad = 11, and bd = 5.
Oof… that middle term looks difficult to tackle by just guessing and checking. But, if we look at the first and last terms, our situation doesn’t seem so bad after all.
Seeing that ac = 2, we can guess that a and c are 1 and 2 (in some order), as those are the only factors of 2. It does not matter which one we choose to be a or which one we choose to be c, because they are both the leading coefficients of different linear factors and the commutative property of multiplication (a×b = b×a) allows us to swap the order of the factors. Let’s let a = 2 and c = 1.
Now we have
(2x+b)(x+d) = (2) x2+ (b+2d) x + bd.
Using similar reasoning for our last term, bd = 5, we can guess that b and d are 1 and 5 (in some order); however this time it does matter which one is which since we have set our values of a and c.
Testing both …
b=5, d=1: (2x+5)(x+1) = 2x2+7x+5 NOPE
b=1, d=5: (2x+1)(x+5) = 2x2+11x+5 YAY
Yes! After trying b=1 and d=5, we have found the quadratic we were trying to factor! Putting everything together:
2x2+11x+5 = (2x+1)(x+5)
and we are finished!
Higher degree polynomials
Generally, you will not need to factor polynomials with a degree higher than 2 on the AP Calculus exam, however if you ever do, here are some tips.
- Take out common factors from each of the terms
- This is especially helpful if you can take out some power of x and reduce the expression into factors of lower degree polynomials
- Use some “guess-and-check” to find the factors
- This is more helpful when the polynomial’s coefficients are prime or have very few factors
- If the roots of the polynomial are given, you can build the factored form using the fact that if r is a root of a polynomial p(x), then (x-r) must be a factor of p(x)
- This is given by the Factor Theorem and is best for when you are given some graph with discernable roots (also called the x-intercepts)
Geometry and Trigonometry
In AP Calculus, you will need to be very comfortable with basic geometry skills as some topics involve triangles, polygon areas, and volumes.
Here are some things you should know before going into the course.
|Square||s = sidelength||s2|
|Rectangle||l = length|
w = width
|Circle||r = radius||πr2|
|Equilateral triangle||s = sidelength||¼(s2)(sqrt(3))|
|Right triangle||a = leg #1 |
b = leg #2
|Trapezoid||b1 = base #1|
b2 = base #2
h = height
|Sphere||R = radius||(4π/3)R3|
|Cone||r = radius|
h = height
|Cylinder||r = radius|
h = height
|Rectangular prism||l = length|
w = width
h = height
If the lengths of the 2 legs of a right triangle are x and y, and the hypotenuse has length z, then the following relation holds true:
x2 + y2 = z2
Note: Below, “opposite” indicates the side length across from the angle θ, “adjacent” indicates the side length next to angle θ that is not the hypotenuse, and “hypotenuse” indicates the hypotenuse of the right triangle
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent
sin2(θ) + cos2(θ) = 1
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