Question

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

b. Use the linear approximation to estimate the given quantity f(x)=4 eˣ

Answer 1

a. The equation of the line representing the linear approximation to the function
`\( f(x) = 4e^x \)` at the poin
`t \( b \) is \( L(x) = 4e^b \cdot (x - b) + 4e^b \)`

b. Using the linear approximation, the estimated quantity for
`\( f(x) = 4e^x \)` at the point
`\( b \)` is
`\( L(x) = 4e^b \cdot (x - b) + 4e^b \).`

Step-by-step explanation:

a. The linear approximation to a function at a given point \( b \) is given by the equation
`\( L(x) = f(b) + f'(b) \cdot (x - b) \)` , where
`\( f'(b) \)` is the derivative of the function at
`\( b \).` For
`\( f(x) = 4e^x \),` the derivative is
`\( f'(x) = 4e^x \),` and thus,
`\( L(x) = 4e^b \cdot (x - b) + 4e^b \).`

b. The linear approximation is a method used to estimate the value of a function near a specific point by using the equation of the tangent line at that point. In this case, the estimated value
`\( L(x) \) for \( f(x) = 4e^x \)` at the point b is obtained by substituting the values of
`\( f(b) \) and \( f'(b) \)` into the linear approximation formula.

c. In the linear approximation formula
`, \( f(b) \)` is the value of the function at the point
`\( b \),` and
`\( f'(b) \)` is the derivative of the function at that point. The result
`\( L(x) \)` provides a good approximation to the function near
`\( b \).` This method is particularly useful in making quick and reasonable estimates for functions that may be challenging to evaluate directly.

Complete Question:

Write the equation of the line that represents the linear approximation to the following function at the given point
`\( b \).` Use the linear approximation to estimate the given quantity
`\( f(x) = 4e^x \).`