Practice Curve Sketching in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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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.

Showing a random 20 of 50 problems.

Example 1

easy
Find f′′(x)f''(x) for f(x)=x3f(x)=x^3.

Example 2

easy
Find the vertical asymptote of f(x)=1x−3f(x)=\frac{1}{x-3}.

Example 3

medium
Find the inflection points of f(x)=x4−6x2f(x) = x^4 - 6x^2.

Example 4

easy
Is f(x)=x2f(x)=x^2 concave up or concave down?

Example 5

easy
Find the yy-intercept of f(x)=x3−2x+5f(x)=x^3-2x+5.

Example 6

hard
Find the global maximum of f(x)=x2e−xf(x)=x^2 e^{-x} on [0,4][0,4].

Example 7

medium
For f(x)=1x2+1f(x)=\frac{1}{x^2+1}, find the maximum and describe the concavity at the maximum.

Example 8

medium
Find the local extrema of f(x)=x3−12xf(x)=x^3-12x.

Example 9

easy
On what interval is f(x)=x2−4xf(x)=x^2-4x decreasing?

Example 10

challenge
For f(x)=xe−xf(x)=xe^{-x}, find the maximum and the inflection point.

Example 11

medium
Why does f(x)=x3f(x)=x^3 have a critical point at 00 that is not an extremum?

Example 12

medium
Find the vertical and horizontal asymptotes of f(x)=x+1x−3f(x)=\frac{x+1}{x-3}.

Example 13

easy
Find the critical points of f(x)=x2−8x+3f(x)=x^2-8x+3.

Example 14

medium
Find the concave-up interval of f(x)=x3−3xf(x)=x^3-3x.

Example 15

easy
Find f′′(x)f''(x) for f(x)=x4f(x)=x^4.

Example 16

medium
Find the horizontal asymptote of f(x)=2x2+1x2−4f(x)=\frac{2x^2+1}{x^2-4}.

Example 17

easy
Where is f(x)=−x2f(x)=-x^2 decreasing?

Example 18

challenge
For f(x)=x1/3(x−4)f(x)=x^{1/3}(x-4), find the critical points and classify them.

Example 19

medium
Find the absolute maximum of f(x)=x3−3xf(x)=x^3-3x on [0,2][0,2].

Example 20

hard
Find the slant (oblique) asymptote of f(x)=x2+1xf(x)=\frac{x^2+1}{x}.
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