What Ap Calc Bc Formulas Should I Put on Calculator

Which Formulas Should You Memorize?

For the AP Calculus BC exam, you'll need to memorize key formulas that are frequently tested. These formulas are typically those that appear most often in multiple-choice and free-response questions. Here are the categories of formulas you should focus on memorizing:

Differentiation Formulas

Memorize the basic differentiation rules:

• Power Rule: \(\frac{d}{dx}[x^n] = n x^{n-1}\)
• Sum/Difference Rule: \(\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\)
• Product Rule: \(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\)
• Quotient Rule: \(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\)
• Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

Integration Formulas

Memorize the basic integration rules and common antiderivatives:

• Power Rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) (for \(n \neq -1\))
• Sum/Difference Rule: \(\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx\)
• Substitution Rule: \(\int f(g(x))g'(x) \, dx = \int f(u) \, du\) where \(u = g(x)\)

Key Theorems

Memorize these important theorems:

• Fundamental Theorem of Calculus
• Mean Value Theorem
• Intermediate Value Theorem
• Rolle's Theorem

Series and Sequences

For the BC portion of the exam, memorize these series formulas:

• Geometric Series: \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}\) (for \(|r| < 1\)) and \(\sum_{n=0}^{N} ar^n = \frac{a(1 - r^{N+1})}{1 - r}\)
• Taylor Series: \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n\)
• Maclaurin Series: \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n\)

Which Formulas Should Be on Your Calculator?

Your graphing calculator should contain formulas that you won't need to memorize but will be useful for solving problems. Here's what you should store:

Basic Functions

Store these basic functions in your calculator:

• Trigonometric functions: \(\sin(x)\), \(\cos(x)\), \(\tan(x)\), \(\csc(x)\), \(\sec(x)\), \(\cot(x)\)
• Inverse trigonometric functions: \(\sin^{-1}(x)\), \(\cos^{-1}(x)\), \(\tan^{-1}(x)\)
• Hyperbolic functions: \(\sinh(x)\), \(\cosh(x)\), \(\tanh(x)\)
• Exponential and logarithmic functions: \(e^x\), \(\ln(x)\), \(\log_{10}(x)\)

Derivatives

Store these derivative formulas in your calculator:

• Derivatives of trigonometric functions: \(\frac{d}{dx}[\sin(x)] = \cos(x)\), \(\frac{d}{dx}[\cos(x)] = -\sin(x)\), \(\frac{d}{dx}[\tan(x)] = \sec^2(x)\)
• Derivatives of inverse trigonometric functions: \(\frac{d}{dx}[\sin^{-1}(x)] = \frac{1}{\sqrt{1 - x^2}}\), \(\frac{d}{dx}[\cos^{-1}(x)] = -\frac{1}{\sqrt{1 - x^2}}\), \(\frac{d}{dx}[\tan^{-1}(x)] = \frac{1}{1 + x^2}\)
• Derivatives of hyperbolic functions: \(\frac{d}{dx}[\sinh(x)] = \cosh(x)\), \(\frac{d}{dx}[\cosh(x)] = \sinh(x)\), \(\frac{d}{dx}[\tanh(x)] = \text{sech}^2(x)\)

Integrals

Store these integral formulas in your calculator:

• Integrals of trigonometric functions: \(\int \sin(x) \, dx = -\cos(x) + C\), \(\int \cos(x) \, dx = \sin(x) + C\), \(\int \tan(x) \, dx = -\ln|\cos(x)| + C\)
• Integrals of inverse trigonometric functions: \(\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x) + C\), \(\int -\frac{1}{\sqrt{1 - x^2}} \, dx = \cos^{-1}(x) + C\), \(\int \frac{1}{1 + x^2} \, dx = \tan^{-1}(x) + C\)
• Integrals of hyperbolic functions: \(\int \sinh(x) \, dx = \cosh(x) + C\), \(\int \cosh(x) \, dx = \sinh(x) + C\), \(\int \text{sech}^2(x) \, dx = \tanh(x) + C\)

Other Useful Formulas

Store these additional formulas that are useful but not typically memorized:

• Area between curves: \(A = \int_{a}^{b} [f(x) - g(x)] \, dx\)
• Volume of revolution: \(V = \pi \int_{a}^{b} [f(x)]^2 \, dx\) (around x-axis)
• Average value of a function: \(\frac{1}{b - a} \int_{a}^{b} f(x) \, dx\)

AP Calc BC Formula Checklist

Use this checklist to organize your memorization and calculator setup:

Example Problems

Here are some example problems that demonstrate how to apply these formulas:

Differentiation Problem

Find the derivative of \(f(x) = x^3 \sin(x)\).

Integration Problem

Evaluate the integral \(\int x \cos(x) \, dx\).

Area Between Curves

Find the area between the curves \(y = x^2\) and \(y = 2x\) from \(x = 0\) to \(x = 2\).