Formulas For Honors Functions/Calc A Exam

$$\frac{d}{dx} [c \cdot u]$$

$$c \cdot u'$$

$$\frac{d}{dx} [u \pm v]$$

$$u' \pm v'$$

$$\frac{d}{dx} [u \cdot v]$$

$$u'v + uv'$$

$$\frac{d}{dx} [\frac{u}{v}]$$

$$\frac{v \cdot u' - u \cdot v'}{v^2}$$

$$\frac{d}{dx} [u^n]$$

$$nu^{n-1}\cdot u^{\prime}$$

$$\frac{d}{dx} [e^u]$$

$$e^u u'$$

$$\frac{d}{dx} [x]$$

$$1$$

$$\frac{d}{dx} [c]$$

$$0$$

$$\frac{d}{dx} [c^u]$$

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

$$\frac{d}{dx} [\sin(u)]$$

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

$$\frac{d}{dx} [\cos(u)]$$

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

$$\frac{d}{dx} [\tan(u)]$$

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

$$\frac{d}{dx} [\cot(u)]$$

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

$$\frac{d}{dx} [\sec(u)]$$

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

$$\frac{d}{dx} [\csc(u)]$$

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

$$\frac{d}{dx} [\ln(u)]$$

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

$$\frac{d}{dx} [\log_c(u)]$$

$$\frac{1}{u \cdot \ln(c)} \cdot u'$$

$g'(x)$ for Inverses

$\frac{1}{f'(g(x))}$ if $f$ and $g$ are inverse functions

Determining Continuty

1. f(a) is exists

2. lim x→a [(f(x)] exists → proven by left and right limits being equal ← if this is not true then in a non-removable discontinuity, either a jump (most common), infinite (VA), or oscillating discontinuity

3. f(a) = lim x→ a [f(x)] ← if only this one is not true then is removable discontinuity (hole or displaced point)

Identities

$\lim_{x\to0}\frac{\left(\sin x\right)}{x}=1$ and $\lim_{x\to0}\frac{\left(1-\cos x\right)}{x}=0$

Limits at Infinity

multiply by 1/x^n, n is the highest power

Intermediate Value Theorem

if smth is a polynomial function that is continuous on the interval [a, b], then for any value L between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = L. Useful for determining if theres a zero