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Cogito

Calculus · Chapter 5 · Lesson 3

Practice Worksheet

Linear Approximation

Use the tangent line instead of the curve.

12 problems · about 21 minutes · CHA-3.F

NameDate

Figure — use these to answer the problems

Look at how closely the tangent hugs the curve near the point, and how quickly it drifts away.
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y = 1√x + 0
  • Point(4, 2)
  • Slope of the tangent0.25
a, coefficient1
Point of tangency4
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. L(x) = 4 + 0.5(x − 16). What is L(16.4)?

    Answer

  2. Why does the estimate worsen far from the base point?

    • The curve bends away from its tangent line.
    • Rounding errors accumulate.

  3. L(x) = 3 + 2(x − 1). What is L(1.5)?

    Answer

  4. f(9) = 3 and f′(9) = 1/6. Estimate f(9.6).

    Answer

  5. f(x) = √x. What is f(25)?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A curve is concave up. Is the estimate high or low? 1 high, 2 low.

    Answer

  2. L(x) = 2 + 0.25(x − 4). What is L(4)?

    Answer

  3. dy = f′(x) dx. With f′(x) = 6 and dx = 0.5, what is dy?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Which choices make a linear approximation more accurate?

    • Choosing a base point close to the target
    • A curve that is nearly straight near the point
    • Moving far from the base point
    • A sharply bending curve
    • Tick every box that applies.

  2. L(x) = 5 + 3(x − 2). What is L(2.1)?

    Answer

  3. The Estimate

    L(x) = 2 + 0.25(x − 4). What is L(4.2)?

    Answer

  4. The Direction

    A curve is concave down. Is the tangent estimate too high or too low? 1 high, 2 low.

    Answer