Unit 1: Limits & Derivatives

Definition of a limit

• describe behaviour of a function near x = a

Conditions for the two-sided limit $\lim_{x \to a} f(x)$ to exist

1. The left-hand limit exists: $\lim_{x \to a^-} f(x)$.

2. The right-hand limit exists: $\lim_{x \to a^+} f(x)$

3. Both one-sided limits are equal: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$

Two special trigonometric limits as $x \to 0$

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ \

$$\lim_{x \to 0} \frac{\cos(x) - 1}{x} = 0$$

Definition of a derivative

• IROC of y with respect to x

• slope of tangent line to a curve at a specific point —> therefore can be found with inputing slope of tangent line with coordinates OR limits

Ways to find derivative using limits

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

What is AROC

$$\text{AROC} = \frac{f(b) - f(a)}{b - a}$$

• Aroc = slope of secant line

Symmetric Difference Quotient (SDQ)

• SDQ is used to estimate the instantaneous rate of change or derivative at $a$

• $\text{SDQ} = \frac{f(a+h) - f(a-h)}{2h}$, works by seeing what 2 numbers a is equally between

Graphing derivative functions

1. If f’(x) has an x incpt at x = a, then f(x) has horizontal tangent at x=a

2. Then start paying attention to axes;

• if f(x) has a positive slope, f’(x) is on the positive x-axes

• if f(x) has a negative slope, f’(x) is on the negative x-axes

• Polynomials —> one lesser degree

Rectilinear motion: condition for moving away from vs. toward the origin

• An object moves away from the origin when position $s(t)$ and velocity $v(t)$ have the same sign

• Moves toward the origin when $s(t)$ and $v(t)$ have opposite signs.

Rectilinear motion: condition for speeding up vs. slowing down

• Speed increases when velocity $v(t)$ and acceleration $a(t)$ share the same sign

• Speed decreases when $v(t)$ and $a(t)$ have opposite signs.

Rectilinear motion: formulas for position, velocity, and acceleration

position: s(t)

velocity: s’(t)

acceleration s’’(t)

Rectilinear motion: formulas displacement

• displacement = change in position

• displacement = s1 - s2

Rectilinear motion: formlas foor instant speed at t1

instant speed at t1 = |v(t1)|

Rectilinear motion: formula for average acceleration over [t1-t2]

Product Rule for differentiation

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Quotient Rule for differentiation

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$

Chain Rule for differentiation

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ or $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Derivatives of $\sin(x)$ and $\cos(x)$

$\frac{d}{dx}[\sin(x)] = \cos(x)$ and $\frac{d}{dx}[\cos(x)] = -\sin(x)$

Derivatives of $e^x$ and $\ln(x)$

$\frac{d}{dx}[e^x] = e^x$ and $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$