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Linear Approximation worksheets

Construct exact linearizations for polynomial, exponential and trigonometric functions, then estimate nearby values to four decimal places.

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Construct L(x) = f(a) + f'(a)(x - a), evaluate L(b) exactly, then approximate f(b) to four decimal places.

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Paper sizes and printing help · More calculus activities

Turn a tangent line into a nearby estimate

Students calculate the exact linear model at a nearby target, then round only the final estimate of the original function.

Learning objective

What this activity builds

Use a tangent-line model to approximate a function value near a known base point.

Worked example

Use the linearization of f(x) = x^2 at a = 2 to estimate f(2.01).

L(x) = 4 + 4(x - 2), so L(2.01) = 4.04 and f(2.01) is approximately 4.0400.

The exact model value L(b) is used as an approximation to the nearby function value f(b).

Teaching notes

Make the practice count

  • Keep f(a), f'(a), a, b and L(b) exact until the last step.
  • Retain the sign of h = b - a.
  • Treat stationary points as constant linear models.

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Choose 6, 8, or 10 problems and any combination of polynomial, exponential, or trigonometric functions. Exact model values are kept separate from four-decimal estimates.

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Math SheetNameScoreTeacherDateLinear ApproximationConstruct L(x) = f(a) + f'(a)(x - a), then use L(b) to estimate f(b). Round only the final estimate to four decimal places.1. Construct L(x), then estimate f(b).f(x) =-x2+3a =0b =-120L(x) =Exact L(b) =f(b) approx.2. Construct L(x), then estimate f(b).f(x) =2·exa =-1b =-1920L(x) =Exact L(b) =f(b) approx.3. Construct L(x), then estimate f(b).f(x) =12·sin-12·x-1a =-1·b =-+120L(x) =Exact L(b) =f(b) approx.4. Construct L(x), then estimate f(b).f(x) =-3·x+x3-3a =1b =1920L(x) =Exact L(b) =f(b) approx.Exact L(b) is the model value; the decimal estimates f(b). Trigonometric angles use radians.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateLinear ApproximationConstruct L(x) = f(a) + f'(a)(x - a), then use L(b) to estimate f(b). Round only the final estimate to four decimal places.5. Construct L(x), then estimate f(b).f(x) =-3·2-3·x-2+1a =-1b =-2120L(x) =Exact L(b) =f(b) approx.6. Construct L(x), then estimate f(b).f(x) =2·cos12·x-2a =0b =-150L(x) =Exact L(b) =f(b) approx.7. Construct L(x), then estimate f(b).f(x) =3·x-2·x2+3·x3+2·x4-1a =2b =10150L(x) =Exact L(b) =f(b) approx.8. Construct L(x), then estimate f(b).f(x) =3·32·x-1a =1b =2120L(x) =Exact L(b) =f(b) approx.Exact L(b) is the model value; the decimal estimates f(b). Trigonometric angles use radians.math-sheet.comPage 2 of 2