We can use differentials (which is the same as linear
approximation) to estimate some complicated functions.
Example 1:
Estimate √4.036 without a calculator.
Solution: We are given x and f and we choose
a. We need a value of a for
which we can compute f and f′, and which is close to our
given x. We are looking at the function
f(x)=√x, x=4.036 and we choose a=4.
Notice that f′(x)=12√x.
Since f(a)=2 and f′(a)=14, we can
estimate
f(x)≈L(x)=f(a)+f′(a)(x−a)=2+.00364=2.009.
This is an extremely accurate approximation.
DO: check this
approximation against the approximation your calculator gives.
Example 2: Estimate e.03 without a calculator.
Solution: We are given x and f, and we choose
a. Again, we need a value of a
for which we can compute f and f′, and which is close to our
given x. This time f(x)=ex, f′(x)=ex,
x=.03 and we let a=0. Then
f(x)≈L(x)=f(a)+f′(a)(x−a)=1+1(.03)=1.03.
This is also an accurate approximation. DO: check this
approximation against the approximation your calculator gives.