AP Calculus AB and BC Memory Sheet

Flashcards covering AP Calculus AB and BC memory formulas, including trigonometric identities, limits, continuity rules, differentiation rules, integration techniques, and Euler's method.

What is the double angle identity for $\sin(2x)$?

$$2\sin(x)\cos(x)$$

What are the three possible expressions for the double angle identity $\cos(2x)$?

$1-2\sin^2(x)$, $\cos^2(x)-\sin^2(x)$, and $2\cos^2(x)-1$

What is the value of the Pythagorean identity $\sin^2(x) + \cos^2(x)$?

$$1$$

According to exponent rules, what does $(a^m)^n$ simplify to?

$$a^{mn}$$

According to exponent rules, what does $a^M a^N$ simplify to?

$$a^{M+N}$$

What is the value of $a^0$?

$$1$$

What is the value of $\ln(e)$?

$$1$$

What is the value of $\ln(1)$?

$$0$$

How do you find the zeros of $f(x)$?

Set $f(x) = 0$

How do you find the intersection of $f(x)$ and $g(x)$?

Set $f(x) = g(x)$ or $f(x) - g(x) = 0$

When evaluating $\lim_{x \to a} \frac{f(x)}{g(x)}$, what should you do first?

Substitute in $a$

When evaluating $\lim_{x \to a} \frac{f(x)}{g(x)}$, what is the result if the numerator is infinity?

The limit is infinity

When evaluating $\lim_{x \to a} \frac{f(x)}{g(x)}$, what is the result if the denominator is infinity?

The limit is $0$

When evaluating $\lim_{x \to a} \frac{f(x)}{g(x)}$, what is the result if the numerator is $0$?

The limit is $0$

When evaluating $\lim_{x \to a} \frac{f(x)}{g(x)}$, what is the result if the denominator is $0$?

The limit is infinity

When does a function have a jump discontinuity?

If the left-hand and right-hand limits exist but are not equal to each other.

When does a function have an infinite discontinuity?

When the left-hand limit, right-hand limit, or both approach infinity or negative infinity.

When does a function have a removable discontinuity at $x = c$?

If the limit of the function exists as $x$ approaches $c$, but the function is either not defined at $c$ or is defined at $c$ with a different value.

What must be shown to prove that $f(x)$ is continuous at $x = a$?

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$$

What is the limit condition for a vertical asymptote at $x = a$?

$$\lim_{x \to a} f(x) = \pm\infty$$

What are the limit conditions for horizontal asymptotes?

$\lim_{x \to \infty} f(x) = a$ or $\lim_{x \to -\infty} f(x) = a$

What are the conditions for the Intermediate Value Theorem (IVT) for the function $f(x)$ from $x = c$ to $x = d$?

The function is continuous on $[c, d]$ and $f(c) \neq f(d)$

What is the limit definition of a derivative?

$$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

Given some $f(x)$ and some value $x = a$, what will $f(a)$ and $f'(a)$ generate?

a) $f(a)$ generates the $y$-value; b) $f'(a)$ generates the slope or $m$

What is the derivative $\frac{d}{dx}[\sin(u)]$?

$$\cos(u) \cdot u'$$

What is the derivative $\frac{d}{dx}[\cos(u)]$?

$$-\sin(u) \cdot u'$$

What is the derivative $\frac{d}{dx}[\ln(u)]$?

$\frac{u'}{u}$ or $\frac{1}{u}$

What is the derivative $\frac{d}{dx}[e^u]$?

$$e^u \cdot u'$$

What is the derivative $\frac{d}{dx}[a^u]$?

$$a^u \ln(a) \cdot u'$$

What is the product rule derivative $\frac{d}{dx}[f(x) \cdot s(x)]$?

$$f'(x)s(x) + s'(x)f(x)$$

What is the quotient rule derivative $\frac{d}{dx}\left[\frac{t(x)}{b(x)}\right]$?

$$\frac{t'(x)b(x)-b'(x)t(x)}{(b(x))^2}$$

What is the derivative $\frac{d}{dx}[\tan(u)]$?

$$\sec^2(u) \cdot u'$$

What is the derivative $\frac{d}{dx}[\sec(u)]$?

$$\sec(u)\tan(u) \cdot u'$$

What is the derivative $\frac{d}{dx}[\csc(u)]$?

$$-\csc(u)\cot(u) \cdot u'$$

What is the derivative $\frac{d}{dx}[\cot(u)]$?

$$-\csc^2(u) \cdot u'$$

What is the chain rule derivative $\frac{d}{dx}[h(g(x))]$?

$$h'(g(x)) \cdot g'(x)$$

What is the derivative $\frac{d}{dx}[\sin^{-1}(u)]$?

$$\frac{1}{\sqrt{1-u^2}} \cdot u'$$

What is the derivative $\frac{d}{dx}[\cos^{-1}(u)]$?

$$\frac{-u'}{\sqrt{1-u^2}}$$

What is the derivative $\frac{d}{dx}[\cot^{-1}(u)]$?

$$\frac{-u'}{u^2+1}$$

Write both ways an inverse trig function can be written for tan.

$\arctan(x)$ and $\tan^{-1}(x)$

What is the derivative $\frac{d}{dx}[f'(x)]$?

$$f''(x)$$

What is the equation of the tangent line to $f(x)$ at $x = a$?

$$y-f(a)=f'(a)(x-a)$$

What is the special limit $\lim_{x \to 0} \frac{\sin(x)}{x}$?

$$1$$

What is the special limit $\lim_{x \to 0} \frac{1-\cos(x)}{x}$?

$$0$$

What is the special limit $\lim_{x \to 0} (1+x)^{\frac{1}{x}}$?

$$e$$

For $\theta = 0$, what are the values of $\tan(\theta)$ and $\csc(\theta)$?

$\tan(0) = 0$ and $\csc(0) = \text{undefined}$

For $\theta = -7\pi$, what are the values of $\sec(\theta)$, $\tan(\theta)$, and $\csc(\theta)$?

$\sec(-7\pi) = -1$, $\tan(-7\pi) = 0$, and $\csc(-7\pi) = \text{undefined}$