Linear Approximation of the following function.

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.

c. Compute the percent error in the​ approximation,100• approximation−exact/ exact, where the exact value is given by a calculator.

f(x)=11−3x2 at a=2​; ​f(2.1​)

.

​a. L(x)=______________

b. Therefore, using the linear approximation ​f(2.1​)≈_____

c. ​Therefore, the percent error in the approximation is approximately _____​%. rounding to three decimal places.

Answer 1

f(x) = 11-3x^2

f(2.1) = 11-3(2.1)^2 = 11 -3(4.41) + 11-13.23 = -2.23

f(2.1) = -2.23 exactly

the linear approximation is

L(x) = f(a)+f'(a)(x-a)

L(2.1) = f(2) +(-12)(2.1-2)

= -1 -1.2

= -2.2= the linear approximation

-2.2 is close to -2.23

it's off by 0.03

0.03/2.23 = .013453 = 1.345% error approximately, rounded to 3 decimal places

graphically, f(x) is a downward opening parabola

the point (2.1, -2.23) is where a tangent line touches the parabola

with the tangent line as y=-12x +23