C15 · Unit C4 Contextual derivatives
Local linearity and approximation error
CoreReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 4.6
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
The tangent model at a is L(x) equals f(a)+f minutes (a)(x minus a). It is useful near a when f is differentiable. If f is concave up on the interval between a and x, the tangent lies below the graph; concave down gives an overestimate. A linear model extrapolated far away need not remain accurate or physically meaningful.
Worked example
To estimate square root of 4.1, take f(x) equals square root of x and a equals 4. Then f(4) equals 2 and f minutes (4) equals 1 over 4, so L(4.1) equals 2.025. Since f seconds (x) equals minus 1/(4x^(3 over 2))<0 for x>0, this is an overestimate.
Explore the mathematics
Local linearity and approximation error
Bring two points together and compare secant and tangent slopes.
Scroll sideways to see the full diagram.
Predict before continuing
Predict how the secant changes as the two points get closer.
Read the complete explanation
- Predict how the secant changes as the two points get closer.
- The orange secant joins two points: its slope is Δy/Δx.
- The green tangent has slope 2. Reduce h to see the secant approach it.
- The derivative is a limit, not substitution of h=0 into a quotient with zero denominator. Here f′(1)=2.
The tangent L(x)=2+(x−4)/4 at x=4 lies above √x because √x is concave down. At 4.1, L=2.025 is a local overestimate.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
The increment is x minus a, not x. A tangent approximation should use a nearby input where exact values are easy.
Investigate and explain
Compare actual and tangent-predicted values at increasing distances from the contact point. Describe a usable local interval.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.