Practice Curve Sketching in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Using the first and second derivatives to determine a function's behavior: intervals of increase/decrease, local maxima/minima, concavity (up/down), and inflection points, then combining this information to sketch an accurate graph.
The first derivative tells you whether the function goes up or down (like reading a speedometer). The second derivative tells you whether it's speeding up or slowing down (like reading an accelerometer). Together, they give you a complete picture of the curve's shape.
Example 1
easySketch f(x) = x^3 - 3x^2: find critical points, monotonicity, concavity, and inflection points.
Example 2
hardSketch f(x) = \dfrac{x^2}{x^2-1}: find domain, asymptotes, critical points, and concavity.
Example 3
easyFor f(x) = x^4 - 4x^3, find and classify all critical points.
Example 4
mediumFind the inflection points of f(x) = x^4 - 6x^2.