A. Write the equation of the line that represents the linear approximation to the following function at the given point a.
b. Graph the function and the linear approximation.
c. Use the linear approx. to estimate the given fxn value.
d. Compute the percent error in your approximation, 100*Iapprox-exactI/IexactI, where the exact value is given by a calculator.

f(x)=4-x^2 at a=1, f(1.1)

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!!!