The Limit as the Instantaneous Rate of Change or Slope of the Tangent Line

The limit represents the instantaneous rate of change of a function, equivalent to the slope of the tangent line at a point. Key concept: As x approaches a specific value, f(x) approaches a specific value L.

Applying Factoring to Evaluate Limits of Rational Functions

When direct substitution yields the indeterminate form 0/0:

1. Factor numerator and denominator

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

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$f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$ This represents the slope of the tangent line at point x.

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

Combine chain rule with other differentiation rules:

• $(e^{f(x)})' = e^{f(x)} \cdot f'(x)$
• $(\sin(f(x)))' = \cos(f(x)) \cdot f'(x)$

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• Position s(t): Location relative to origin
• Velocity v(t): $v(t) = s'(t)$ = instantaneous rate of change of position
• Acceleration a(t): $a(t) = v'(t) = s''(t)$ = instantaneous rate of change of velocity

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• Positive velocity: moving in positive direction (right/up)
• Negative velocity: moving in negative direction (left/down)
• Speeding up: velocity and acceleration have same sign

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Special case of MVT: If f(a) = f(b) = 0 and f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that f'(c) = 0.

If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$ Geometric meaning: Slope of tangent equals slope of secant line at some point.

• Left Riemann Sum (LRAM): $\sum_{i=1}^n f(x_{i-1})\Delta x$
• Right Riemann Sum (RRAM): $\sum_{i=1}^n f(x_i)\Delta x$
• Midpoint Riemann Sum (MRAM): $\sum_{i=1}^n f(\bar{x}_i)\Delta x$ where $\bar{x}_i = \frac{x_{i-1}+x_i}{2}$

$\int_a^b f(x)dx \approx \frac{\Delta x}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]$ More accurate than Riemann sums for smooth functions.

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

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

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Graphical representation showing $\frac{dy}{dx}$ at various points (x, y).

• Each small line segment has slope = dy/dx at that point
• Solution curves follow the slope field pattern

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

s(t) = ∫v(t)dt

$s(t) = \int v(t)dt$ Position is the integral of velocity.