Five Things You Must Know in AP Calculus

Success in AP Calculus is highly valued in college admissions because it demonstrates a student’s readiness for advanced math courses and their ability to excel in challenging academic environments. A strong performance in AP Calculus signals to admissions committees that you possess crucial skills necessary for success in college-level STEM fields.

The College Board outlines the course structure by organizing it into three Big Ideas.

1. Big idea 1: Change (CHA)
2. Big idea 2: Limits (LIM)
3. Big idea 3: Analysis of Functions (FUN)

And here’s a breakdown of the AP Calculus curriculum into five big categories:

1. Limits and Continuity
2. Differentiation
3. Integration and Accumulation
4. Applications of Derivatives and Integrals
5. Series and Sequences

1. Limits and Continuity

This category covers the fundamental concepts of limits and continuity of functions. Students learn about the behavior of functions as they approach certain values, understand the definition of limits, and explore different types of discontinuities.

Sample question

For what value of the constant \(c\) is the function \(f\left(x\right)\) continuous on \(\left(-\infty,\,\infty\right)\) ?

\(f\left(x\right)=\begin{cases} cx^2+2x\,\,\, &\mbox{if}& \,\,x<2 \\ x^3-cx\,\,\, &\mbox{if}& \,\,x\geq 2 \end{cases}\)

Answer

\(c=\displaystyle\frac{2}{3}\)

2. Differentiation

Differentiation is a central topic in calculus and involves finding derivatives of functions. This category includes techniques such as the power rule, product rule, quotient rule, chain rule, and implicit differentiation.

Sample question

Find \(y^{\prime\prime} \) if \(x^4+y^4=16\).

Answer

\(y^{\prime\prime}=-\displaystyle\frac{248x^2}{y^7}\)

3. Integration and Accumulation

Integration is the process of finding antiderivatives and calculating definite and indefinite integrals. This category covers techniques of integration, such as substitution, integration by parts, trigonometric integrals, and partial fractions.

Sample question

Find \(\displaystyle\int\frac{x^4-2x^2+4x+1}{x^3-x^2-x+1}\).

Answer

\(\displaystyle\frac{x^2}{2}+x-\frac{2}{x-1}+\ln\left|\frac{x-1}{x+1}\right|+K\) where \(K\) is an integration constant.

4. Applications of Derivatives and Integrals

This category focuses on applying differentiation and integration to solve real-world problems. Topics include optimization, modeling with differential equations, related rates, motion along a line, area/volume problems, parametric equations, polar coordinates, and vector-valued functions.

Sample question

The region \(\mathcal{R}\) enclosed by the curves \(y=x\) and \(y=x^2\) is rotated about the \(x-\)axis. Find the volume of the resulting solid.

Answer

\(\displaystyle\frac{2\pi}{15}\)

Sample question

Find the area of the region that lies inside the circle \(r=3\sin\theta\) and outside the cardioid \(r=1+\sin\theta\).

Answer

\(\pi\)

5. Series and Sequences

Series and sequences involve the study of infinite sums and the behavior of sequences of numbers. This category covers convergence and divergence of series, tests for convergence, Taylor and Maclaurin series, and applications of series such as approximating functions..

Sample question

Find the Maclaurin series for (a) \(f\left(x\right)=x\cos x\) and (b) \(f\left(x\right)=\ln\left(1+3x^2\right)\).

Answer

(a) \(\displaystyle\sum_{n=0}^{\infty}\left(-1\right)^n\frac{x^{2n+1}}{\left(2n\right)!}\)

(b) \(\displaystyle\sum_{n=1}^{\infty}\left(-1\right)^{n-1}\frac{3^n\cdot x^{2n}}{n}\)