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Linear Approximation

Learn linear approximation, differentials.

Lesson 3 of 32 parts Practice Flashcards

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1

Linear Approximation

Objective: Explain linear approximation, apply the stated method, and check its conditions.

Near a differentiable input a, f(x)≈f(a)+f′(a)(x−a). The tangent model retains first-order change and omits higher-order terms. Its accuracy depends on the step size and the function’s curvature; it is an approximation, not an identity. A tangent estimate should not be presented as an exact value without justification.

Formulas

$L(x)=f(a)+f′(a)(x−a)$

Worked examples

Try itUse the tangent approximation to f(x)=x² at x=7 to estimate f(7+1/10).Show answer

Step 1: L(x)=49+14(x−7). Step 2: The input change is 1/10. Step 3: L(7+1/10)=252/5; this is an approximation.

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1

Linear Approximation

Objective: Explain linear approximation, apply the stated method, and check its conditions.

Near a differentiable input a, f(x)≈f(a)+f′(a)(x−a). The tangent model retains first-order change and omits higher-order terms. Its accuracy depends on the step size and the function’s curvature; it is an approximation, not an identity. A tangent estimate should not be presented as an exact value without justification.

Formulas

$L(x)=f(a)+f′(a)(x−a)$

Worked examples

Try itUse the tangent approximation to f(x)=x² at x=7 to estimate f(7+1/10).Show answer

Step 1: L(x)=49+14(x−7). Step 2: The input change is 1/10. Step 3: L(7+1/10)=252/5; this is an approximation.

2

Differentials

Objective: Explain differentials, apply the stated method, and check its conditions.

For y=f(x), the differential dy=f′(x)dx represents the tangent-model output change for an input increment dx. The actual change is Δy=f(x+dx)−f(x). They agree only in special cases; for small steps dy can approximate Δy. dy and the actual Δy are generally different.

Formulas

$dy=f′(x)dx$

Worked examples

Try itFor y=6x² at x=7, use dx=1/10. Find the differential dy.Show answer

Step 1: Use dy=f′(x)dx. Step 2: f′(7)=84; multiply by 1/10. Step 3: dy=42/5; it approximates the output change.

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