Question

a. Write the equation of the line that represents the linear approximation to the following function at the given point a. b. Use the linear approximation to estimate the given quantity. approximation - exact| c. Compute the percent error in the approximation, 100. where the exact value is given by a calculator. exact f(x) = 2 eat a = 0; f(0.03) a. L(x)= b. Using the linear approximation, f(0.03) (Type an integer or a decimal.) c. The percent error in the approximation is 1%. (Round to four decimal places as needed.)

Answer 1

a.
The linear approximation to the given function at the point a = 0 is given by the linear function L : $\mathbb{R}$ $\rightarrow$$\mathbb{R}$ defined by,
L(x) = f(0) + f'(0)x. Now, the derivative of f at 0 is 2e⁰ = 2.
Thus, the equation of the line that represents the linear approximation to the function f at a = 0 is
L(x) = 2 + 2x.

b.
Now, f(0.03) $\approx$ L(0.03) = 2 + 2 x 0.03 = 2.06

c.
With the aid of a calculator, f(0.03) = 2e^0.03 = 2.06091
Thus, the percent error in the approximation is given by,

2.06 -2.06091|| 100 本 2006 0.0442