A limit describes the value that a function approaches as the input approaches a certain point. Written as: $\lim_{x \to c} f(x) = L$ The limit can exist even if the function is undefined at point c.

Instantaneous velocity is the limit of average velocity as the time interval approaches zero: $v(t_0) = \lim_{\Delta t \to 0} \frac{s(t_0 + \Delta t) - s(t_0)}{\Delta t}$

Average rate of change over [a, b]: $\frac{f(b) - f(a)}{b - a}$

Instantaneous rate of change at x = a: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ Alternative form:

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$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ Using Leibniz notation: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Differentiation of f(g(x))

Example: $\frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x$ Example: $\frac{d}{dx}((3x+1)^5) = 5(3x+1)^4 \cdot 3 = 15(3x+1)^4$

Relationships: v(t) = s'(t)

Velocity is the derivative of position: $v(t) = s'(t) = \frac{ds}{dt}$

Relationships: a(t) = v'(t) = s''(t)

Acceleration is the derivative of velocity, second derivative of position: $a(t) = v'(t) = s''(t) = \frac{d^2s}{dt^2}$

If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$ Geometrically: There's a point where tangent is parallel to secant line.

Special case of MVT: If f(a) = f(b) = 0, then there exists c ∈ (a, b) where f'(c) = 0.

Left sum, right sum, midpoint sum, trapezoidal sum.

Approximation of area under curve using rectangles: $\int_a^b f(x)dx \approx \sum_{i=1}^n f(x_i^*) \Delta x$ where $\Delta x = \frac{b-a}{n}$

To verify y = g(x) is a solution to differential equation:

1. Find y', y'', etc.
2. Substitute into differential equation

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Graphical representation showing $\frac{dy}{dx}$ at various points. Each small segment shows slope of solution at that point.

Average Value Formula: (1/(b-a))∫abf(x)dx\int_a^b f(x)dx∫ab​f(x)dx

$f_{avg} = \frac{1}{b-a}\int_a^b f(x)dx$

s(t) = ∫v(t)dt\int v(t)dt∫v(t)dt

$s(t) = \int v(t)dt$

Parametric Equations x(t), y(t)

Curve defined by: x = f(t), y = g(t), t ∈ [a, b]

dy/dx = (dy/dt)/(dx/dt)

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$

• Sequence: Ordered list of numbers {an}
• Series: Sum of sequence terms: $\sum_{n=1}^\infty a_n$

Partial sums: $S_N = \sum_{n=1}^N a_n$ Series converges if $\lim_{N \to \infty} S_N$ exists (finite).