T
Tutorial • 81
Applications of Differentiation
3.8 Tangent Line Approximation and Differentials
The line tangent to a curve at a point is the line that best approximates the curve near that
point. So, we use the tangent line at the point (c,f(c)) as an approximation to the curve
y = f{x) when x is near c.
An equation for the tangent line at the point (c,f(c)) is given by
y-m = f(cXx-c) or y = m+fXcXx-c)
and the approximation
f(x)*f{c)+f'{c)(x-c)
is called the tangent line approximation of / at c.
y=X*).
y=Ac)+f(c)(x-c)
The linear function
K*) = /(c)+/'(c)(*-c)
is called the linearization of / at c.
Definition of Differentials
Let y = f(x) be a differentiable function. The differential dx is an independent variable.
The differential dy is
dy = f(x)dx.
Estimating with Differentials
Differentials can be used to approximate function values.
f(x+Ax) « f(x)+dy * f(x)+f'{x)dx
Example 1 o (a) Find the tangent line approximation of f(x) = -Jx-\ at c = 5
and approximate the number >/3.95.
(b) Find the tangent line approximation of f{x) = tan* at c = tt/4
and approximate the number tan 47".
82 • Tutorial
Chapter 3
Solution d (a)/(x) = >/x^T r> /(5) = >/5-f = 2
/'(*) =
2,/^T
/'(5) =
^ = 2 +41(^-5)
1
3
y=—x+4 4
/ — T«-x+—
! 3
VJf-1
4 4_
>^95=V4.95-1
1
2>/5^T 4
Tangent line approximation
Simplify.
f(x)*f(c)+f(cXx-c)
x = 4.95
*-(4.95)+-=1.9875
4
4
(b) f(x) = tan x => /(f) = tan(f) = 1
4
4
,fl\
„2,*v
/'(*) = sec2* => /'(7) = sec2(^-)
=2
4
4
^ = l + 2(x-^-)
4
Tangent line approximation
'"te-f+1
Simplify.
tan jc « 2* +1
2
f ( x ) ~ fi c ) + f ( c X x - c )
tan 47"* 2(47°-— )180
2
x = 47° and 47° = 47 radian
180
* 1.0698