More Applications of Derivatives 185Example 3
The slope of a function at any point (x,y)is−
x+ 1
y. The point (3, 2) is on the graph of
f. (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:y− 2 =−2(x−3) ory=− 2 x+8.
(b) f(3.1)≈−2(3.1)+ 8 ≈ 1. 8Estimating 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 approximation 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 /^3andf′(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: You must express the angle measurement in radians before applying linear approxi-
mations. 30◦=
π
6
radians and 1◦=
π
180
radians.Letf(x)=sinx,a=
π
6andΔx=
π
180