Question

A. Write the equation of the line that represents the linear approximation to the following function at the given point a.

Answer 1

Answer:

a) f(x) = 4 - x²

The linear approximation of the function at a=1 is

L(x) = 5 - 2x at a = 1

b) The graph of the function and the linear approximation at that point is attached to this solution.

The curve represent the real function,

f(x) = 4 - x²

The straight line represents the linear approximation of the function at a=1.

L(x) = 5 - 2x

The curve and the function evidently cross paths at x=1 and understandably so.

c) Using the linear approximation obtained at a = 1.

f(1.1) = 2.8

Using the actual function, the actual value of f(1.1) = 2.79

d) Percent error = 0.358%

Step-by-step explanation:

f(x) = 4 - x²

a) The linear approximation of the function at the given point is given as

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

f(x) = 4 - x²

a = 1

f(a) = 4 - 1² = 3

f'(x) = -2x

f'(a) = -2(1) = -2

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

L(x) = 3 + (-2)(x - 1)

L(x) = 3 -2x + 2

L(x) = 5 - 2x

L(x) = -2x + 5

f(x) = 4 - x²

L(x) = 5 - 2x at a = 1

b) The graph of the function and the linear approximation at that point is attached to this solution.

The curve represent the real function,

f(x) = 4 - x²

The straight line represents the linear approximation of the function at a=1.

L(x) = 5 - 2x

The curve and the function evidently cross paths at x=1 and understandably so.

c) Use the linear approx. to estimate the given fxn value.

f(1.1)

L(x) = 5 - 2x

L(1.1) = 5 - 2(1.1) = 2.8

Using the function, the actual value of f(1.1) = 4 - 1.1² = 2.79

d) Compute the percent error in your approximation, 100*Iapprox-exactI/IexactI, where the exact value is given by a calculator

Percent error

= 100% × (|approx - exact|)/exact

Approximated value = 2.8

Exact value = 2.79

Percent error = 100% × (2.8-2.79)/2.79

Percent error = 0.358%

Hope this Helps!!!