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Mean Value Theorem worksheets

Find every exact interior value where a derivative equals the secant slope for polynomial, trigonometric, exponential, and logarithmic functions.

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For each function on [a, b], find every c in (a, b) such that f'(c) equals the secant slope. Show endpoint values, the slope, and the derivative equation.

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Match a tangent slope to the whole interval

Each supported function is continuous on the closed interval and differentiable inside it. The worked key evaluates both endpoints, forms the secant slope, and solves the exact derivative equation.

Learning objective

What this activity builds

Evaluate both endpoints, form the secant slope, solve f'(c) = m, and retain every solution strictly inside the stated interval.

Worked example

For f(x) = x^2 on [0, 2], find the MVT value c.

c = 1.

The secant slope is (4 - 0)/(2 - 0) = 2. Since f'(x) = 2x, solving 2c = 2 gives c = 1, which lies in (0, 2).

Teaching notes

Make the practice count

  • Write the open interval before filtering solutions.
  • Keep exact pi, inverse-cosine, logarithm, fraction, and radical forms.
  • A periodic derivative can produce several valid c-values.

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Math SheetNameScoreTeacherDateMean Value TheoremFor each function on [a, b], find every c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a). Show the endpoint values, secant slope, and derivative equation.1. Find every c in (a, b) with f'(c) equal to the secant slope.f(x) =x2+4·x1+4[a, b] = [-1,4]2. Find every c in (a, b) with f'(c) equal to the secant slope.f(x) =x3-6·x2+9·x1[a, b] = [-5,-1]3. Find every c in (a, b) with f'(c) equal to the secant slope.f(x) =-x2-4·x1-3[a, b] = [-5,-1]4. Find every c in (a, b) with f'(c) equal to the secant slope.f(x) =-x3+2·x1+4[a, b] = [1,5]MVT: continuous on [a, b], differentiable on (a, b), and every reported c lies strictly inside.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateMean Value TheoremFor each function on [a, b], find every c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a). Show the endpoint values, secant slope, and derivative equation.5. Find every c in (a, b) with f'(c) equal to the secant slope.f(x) =-2·x2+8·x1-10[a, b] = [1,5]6. Find every c in (a, b) with f'(c) equal to the secant slope.f(x) =x3-6·x2+12·x1-11[a, b] = [-6,2]7. Find every c in (a, b) with f'(c) equal to the secant slope.f(x) =-2·x2+8·x1-6[a, b] = [-6,2]8. Find every c in (a, b) with f'(c) equal to the secant slope.f(x) =x3-3·x2+4·x1+2[a, b] = [-2,6]MVT: continuous on [a, b], differentiable on (a, b), and every reported c lies strictly inside.math-sheet.comPage 2 of 2

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