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Calculus practice

Average rate of change over an interval

Find twelve finite-interval rates for quadratic, cubic, rational and natural exponential functions.

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Find (f(b) - f(a))/(b - a). Round only a final nonterminating result to four decimals.

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Practice with exact answers

Keep fractions, radicals and pi exact. Round only when a question asks for a decimal answer, and check every original domain restriction.

Learning objective

What this activity builds

Evaluate a function at two endpoints and form the exact secant quotient.

Worked example

Find the average rate of change of f(x) = x² + 1 on [1, 3].

(f(3) − f(1)) ÷ (3 − 1) = (10 − 2) ÷ 2 = 4.

Evaluate both endpoints first: f(1) = 2 and f(3) = 10. The output increases by 8 while the input increases by 2, so the secant slope is 4. This is the average over the interval, not the derivative at an endpoint.

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  • Keep exact values through the final step.
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  • Treat trigonometric inputs as radians.
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Math SheetAverage Rate of ChangeFind (f(b) - f(a))/(b - a). Round only final nonterminating values to 4 decimals.1.f(x) = (x^2+1)/(x-2); [5, 8]2.f(x) = 1e^(-1x+0); [1, 3]3.f(x) = -2x^3+x^2+4x-1; [-2, 1]4.f(x) = x^3-2x+3; [-2, 1]5.f(x) = x^2-3x+2; [-2, 1]6.f(x) = -2x^2+4x-5; [-2, 1]math-sheet.comPage 1 of 2
Math SheetAverage Rate of ChangeFind (f(b) - f(a))/(b - a). Round only final nonterminating values to 4 decimals.7.f(x) = 3x^2-6x+1; [-2, 1]8.f(x) = 2e^(3x-2)+1; [1, 3]9.f(x) = 2e^(-1x+0); [1, 3]10.f(x) = -x^3+2x^2+x-3; [-2, 1]11.f(x) = (x^2-6x)/(2x^2-12x+18); [5, 8]12.f(x) = (1/2)/(x^2-4x+4)+(-3)/(x^2); [4, 7]math-sheet.comPage 2 of 2

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