Calculus
Average rate of change over an interval
Find twelve finite-interval rates for quadratic, cubic, rational and natural exponential functions.
Preview and downloadGeneral functions practice
Difference quotient worksheets
Simplify eight difference quotients with polynomial, factored and rational functions. Choose two inputs or an increment, add a formula reminder and print worked answers.
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Use the requested inputs to form the change in output divided by the change in input. Substitute with parentheses, factor the numerator and simplify. Keep all original denominator restrictions.
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Two inputs. One exact rate of change.
Choose the two-point quotient (f(x) − f(a)) ÷ (x − a), or the increment quotient (f(x + h) − f(x)) ÷ h. Each sheet has eight functions: expanded and factored polynomials, an affine fraction, a reciprocal and an affine-over-linear rational function.
Print the formula as a reminder or leave it out for independent practice. Keeping the seed preserves the functions across both choices, so you can compare the two methods on the same examples.
The worked key shows substitution, a factored cancellation step, the simplified result and the original restrictions. For rational functions, both substituted denominators must remain nonzero.
Learning objective
What this activity builds
Build a difference quotient from a function, simplify exactly and preserve the restrictions from the original quotient.
Worked example
If f(x) = x² + 3, simplify (f(x + h) − f(x)) ÷ h.
((x + h)² + 3 − (x² + 3)) ÷ h = (2xh + h²) ÷ h = 2x + h, with h ≠ 0.
Expand the complete squared input before subtracting f(x). The constant terms cancel. Factor h out of the numerator, then cancel it while retaining the original restriction h ≠ 0.
Teaching notes
Make the practice count
- Start with the formula reminder, then hide it when learners can reconstruct the quotient.
- Keep the same seed when comparing the two formula choices; the function definitions stay the same.
- Subtract the entire second function value, including every sign.
- For a rational function, combine the two fractions using a common denominator before canceling.
- Keep x ≠ a or h ≠ 0 even when that factor cancels.
- Also exclude inputs where either original function value is undefined.
- Check a symbolic result with a pair of allowed numerical inputs.
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