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AP Calculus AB: 7.2.2 Using the Tangent Line Approximation Formula

Mathematics14 CardsCreated 3 months ago

These flashcards explain the tangent line approximation formula, which uses the derivative at a known point to approximate function values near that point. The cards cover the process of selecting an easy point, calculating the slope via the derivative, and applying the formula for practical approximations of functions like exponentials, roots, and logarithms.

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Using the Tangent Line Approximation Formula

The tangent line approximation formula is f(x+delta x) = f(x) + f’(x)(delta x)

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Key Terms

Term
Definition

Using the Tangent Line Approximation Formula

The tangent line approximation formula is f(x+delta x) = f(x) + f’(x)(delta x)

note

  • The process of finding a linear approximation can be
    described by a general formula.

  • Use the linear approximation formula to approximate e^2.1.

    e^2.1≈11e^2/10

Use the linear approximation formula to approximate 3√7.9.

√7.9≈239/120

Use the linear approximation formula to approximate ln 1.1.

ln1.1≈1/10

Find the linearization of the function f(x)=√x+8 at a=1.

y=x/6+17/6

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AP Calculus AB: 7.2.2 Using the Tangent Line Approximation Formula
TermDefinition

Using the Tangent Line Approximation Formula

The tangent line approximation formula is f(x+delta x) = f(x) + f’(x)(delta x)

note

  • The process of finding a linear approximation can be
    described by a general formula.

  • When making a linear approximation, you start by finding the equation of the line tangent to the curve at an “easy point.” A point is considered easy if you can evaluate the function at that point. For example, the square root function is easy to evaluate at the number 9.

  • The distance between the easy point and the point you are interested in is the change in x, or x.

  • Use the point-slope form of a line. Notice that the slope of the tangent line is equal to the first derivative of the curve evaluated at the “easy point.”

  • The y-value of this equation tells you the height of the line at that x-point. Remember, this y-value is a good
    approximation for the function at that point. This equation is called the tangent line approximation formula.

  • To use the tangent line approximation formula, start by finding a good easy point and the distance between the easy point and the point you want to approximate.

  • In this example you are asked to approximate the value of the cube root of 7.9. Since you know that the cube root of 8 is 2, you can use that point as your easy point. The signed distance between 8 and 7.9 is –0.1.

  • Find the derivative of the cube root function using the power rule. Evaluate the derivative at the “easy point.” Finally, plug all of that information into the linear approximation formula.

  • The resulting approximation is accurate to 4 decimal places.

Use the linear approximation formula to approximate e^2.1.

e^2.1≈11e^2/10

Use the linear approximation formula to approximate 3√7.9.

√7.9≈239/120

Use the linear approximation formula to approximate ln 1.1.

ln1.1≈1/10

Find the linearization of the function f(x)=√x+8 at a=1.

y=x/6+17/6

What is the largest interval of x for which the linear approximation √1+x≈1+x/2 is accurate to within .2?

(.4−√1.6,.4+√1.6)

Use the linear approximation formula to approximate

3√27.1.

√27.1≈811/270

Use the linear approximation formula to approximate e^3.1.

e3.1≈11e^3/10

Use the linear approximation formula to approximate sec 99π/100

sec99π/100≈−1

Use the linear approximation formula to approximate √16.2.


√16.2≈161/40

Use the linear approximation formula to approximate the root.
√3.9

√3.9≈79/40

Use the linear approximation formula to approximate tan 0.1.

tan.1≈0.1

Use the linear approximation formula to approximate √15.9.

√15.9≈319/80