Calculus- Early Transcendentals, 2021a

184 Applications of Derivatives
Solution. If f (x)=sinx, then f ′ (x)=cosx, and thus the linear approximation of sinx at x = 0is:
L(x)=cos(0)(x − 0)+sin(0)=x.
Thus when x is small this is quite a good approximation and is used frequently by engineers and scientists
to simplify some calculations.
For example you can use your calculator (in radian mode since the derivative of sinx is cosx only in
radian) to see that
sin(0.1)=0.099833416...
and thus L(0.1)=0.1 is a very good and quick approximation without any calculator!
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Exercises for 5.4.1
Exercise 5.4.1 Find the linearization L(x) of f (x)=ln(1 + x) at a = 0. Use this linearization to approximate
f (0.1).
Exercise 5.4.2 Use linear approximation to estimate (1.9) 3 .
Exercise 5.4.3 Show in detail that the linear approximation of sinx atx= 0 is L(x) =x and the linear
approximation of cosx atx= 0 is L(x)=1.
Exercise 5.4.4 Use f (x) = 3√ x + 1 to approximate 3√ 9 by choosing an appropriate point x = a. Are we
over- or under-estimating the value of 3√ 9?Explain.
5.4.2 Differentials
Very much related to linear approximations are the differentials dx and dy, used not to approximate values
of f , but instead the change (or rise) in the values of f .
Definition 5.29: Differentials dx and dy
Let y = f (x) be a differentiable function. We define a new independent variable dx, andanew
dependent variable dy = f ′ (x)dx. Notice that dy is a function both of x (since f ′ (x) is a function of
x) and of dx. We call both dx and dy differentials.
Now fix a point a and let Δx = x − a and Δy = f (x) − f (a). Ifx is near a then Δx is clearly small. If
we set dx = Δx then we obtain
dy = f ′ (a)dx ≈ Δy Δx = Δy.
Δx
Thus, dy can be used to approximate Δy, the actual change in the function f between a and x. This is
exactly the approximation given by the tangent line:
dy = f ′ (a)(x − a)= f ′ (a)(x − a)+ f (a) − f (a)=L(x) − f (a).
While L(x) approximates f (x), dy approximates how f (x) has changed from f (a). Figure 5.9 illustrates
the relationships.