5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 15:14

More Applications of Derivatives 213

Example 3

The slope of a function at any point (x,y)is−

x+ 1

y

. The point (3, 2) is on the graph off.
(a) Write an equation of the line tangent to the graph off atx=3.
(b) Use the tangent line in part (a) to approximatef(3.1).

(a) Lety=f(x), then

dy

dx

=−

x+ 1

y

dy

dx

∣∣

∣∣

x=3,y= 2

=−

3 + 1

2

=− 2.

Equation of tangent line:y− 2 =−2(x−3) ory=− 2 x+8.

(b) f(3.1)≈−2(3.1)+ 8 ≈ 1. 8

Estimating thenth Root of a Number

Another way of expressing the tangent line approximation is:

f(a+Δx)≈ f(a)+ f′(a)Δx, whereΔxis a relatively small value.

Example 1

Find the approximate value of

√

50 using linear approximation.

Usingf(a+Δx)≈ f(a)+f′(a)Δx, letf(x)=

√

x;a=49 andΔx=1.

Thus,f(49+1)≈ f(49)+f′(49)(1)≈

√

49 +

1

2

(49)−^1 /^2 (1)≈ 7 +

1

14

≈ 7 .0714.

Example 2

Find the approximate value of^3

√

62 using linear approximation.

Letf(x)=x^1 /^3 , a=64, Δx=−2. Since f′(x)=

1

3

x−^2 /^3 =

1

3 x^2 /^3

and

f′(64)=

1

3(64)^2 /^3

=

1

48

, you can use f(a+Δx) ≈ f(a)+ f′(a)Δx. Thus, f(62)=

f(64−2)≈f(64)+ f′(64)(−2)≈ 4 +

1

48

(−2)≈ 3 .958.

TIP • Use calculus notations and not calculator syntax, e.g., write

∫

x^2 dx and not

∫

(x∧2, x).

Estimating the Value of a Trigonometric Function of an Angle

Example

Approximate the value of sin 31◦.

Note that you must express the angle measurement in radians before applying linear

approximations. 30◦=

π

6

radians and 1◦=

π

180

radians.

Letf(x)=sinx,a=

π

6

andΔx=

π

180

.