7.2.2 Using the Tangent Line Approximation Formula

Question 1

Using the Tangent Line Approximation Formula

Answer 1

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

Question 2

note

Answer 2

• 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.

Question 3

Use the linear approximation formula to approximate e^2.1.

Answer 3

e^2.1≈11e^2/10

Question 4

Use the linear approximation formula to approximate 3√7.9.

Answer 4

√7.9≈239/120

Question 5

Use the linear approximation formula to approximate ln 1.1.

Answer 5

ln1.1≈1/10

Question 6

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

Answer 6

y=x/6+17/6

Question 7

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

Answer 7

(.4−√1.6,.4+√1.6)

Question 8

Use the linear approximation formula to approximate

3√27.1.

Answer 8

√27.1≈811/270

Question 9

Use the linear approximation formula to approximate e^3.1.

Answer 9

e3.1≈11e^3/10

Question 10

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

Answer 10

sec99π/100≈−1

Question 11

Use the linear approximation formula to approximate √16.2.

Answer 11

√16.2≈161/40

Question 12

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

Answer 12

√3.9≈79/40

Question 13

Use the linear approximation formula to approximate tan 0.1.

Answer 13

tan.1≈0.1

Question 14

Use the linear approximation formula to approximate √15.9.

Answer 14

√15.9≈319/80