Optimization Problems in Calculus: 4 Solved Examples
Solve calculus optimization problems in 6 steps. Worked examples: the largest fenced area, the open box, the cheapest can and the closest point on a curve.
Category: Applications
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Related Rates: Step-by-Step Strategy and Classic Problems
Solve related rates problems in 6 steps. Worked classics: the sliding ladder, the expanding balloon, the draining cone and the moving shadow.
Area Between Two Curves: Step-by-Step Method and Examples
Area between curves = ∫(top − bottom)dx. Find intersection points, handle curves that cross, and integrate with respect to y when easier. Worked examples.
Riemann Sums: Left, Right, Midpoint and Simpson’s Rule
How to compute left, right and midpoint Riemann sums, the trapezoidal rule and Simpson’s rule, with a full worked example and error comparison.
Average Value of a Function: Formula, Meaning and Examples
The average value of f on [a, b] is (1/(b − a))∫f(x)dx. Why the formula works, examples with sin x, x² and eˣ, and the Mean Value Theorem for integrals.
Shell Method for Volume: Formula, Examples and Disk vs Shell
The shell method finds volumes of revolution with V = 2π∫(radius)(height)dx. Step-by-step examples and a clear guide to choosing shells or disks.
Disk and Washer Method: Volume of Revolution Step by Step
Find volumes of solids of revolution with the disk method V = π∫f(x)²dx and the washer method V = π∫(R² − r²)dx. Worked examples and setup tips.
Arc Length Formula in Calculus: Derivation and Examples
The arc length of y = f(x) from a to b is ∫√(1 + [f'(x)]²) dx. Derivation, worked examples (x^(3/2), a circle, a catenary) and parametric curves.
Mean Value Theorem: Statement, Meaning and Worked Examples
The Mean Value Theorem guarantees a point where the instantaneous rate equals the average rate. Hypotheses, Rolle’s theorem, and how to find c step by step.
Linear Approximation (Linearization): Formula and Examples
Linear approximation uses the tangent line L(x) = f(a) + f'(a)(x − a) to estimate values like √4.1 and sin 0.1. Includes error, differentials and examples.