Questions: Use a linear approximation to estimate the following quantity. Choose a value of a that produces a small error and does not require the use of a calculator. e^0.02 The linear approximation of e^0.02 is (Type an exact answer.)

Use a linear approximation to estimate the following quantity. Choose a value of a that produces a small error and does not require the use of a calculator.

e^0.02

The linear approximation of e^0.02 is
(Type an exact answer.)

Transcript text: Use a linear approximation to estimate the following quantity. Choose a value of a that produces a small error and does not require the use of a calculator. \[ e^{0.02} \] The linear approximation of $e^{0.02}$ is $\square$ (Type an exact answer.)

Solution

Solution Steps

To estimate $ e^{0.02} $ using linear approximation, we can use the formula for linear approximation $ f(x) \approx f(a) + f'(a)(x-a) $. Here, $ f(x) = e^x $, and we choose $ a = 0 $ because $ e^0 = 1 $ and the derivative $ f'(x) = e^x $ is also 1 at $ x = 0 $. This choice simplifies the calculation and minimizes error for small $ x $.

Solution Approach

Identify the function $ f(x) = e^x $ and its derivative $ f'(x) = e^x $.
Choose $ a = 0 $ for simplicity, where $ f(a) = e^0 = 1 $ and $ f'(a) = e^0 = 1 $.
Apply the linear approximation formula: $ f(x) \approx f(a) + f'(a)(x-a) $.
Substitute $ x = 0.02 $ and $ a = 0 $ into the formula to estimate $ e^{0.02} $.

Step 1: Define the Function and Derivative

We start with the function $ f(x) = e^x $ and its derivative $ f'(x) = e^x $. For our linear approximation, we will evaluate these at the point $ a = 0 $.

Step 2: Evaluate at $ a = 0 $

At $ a = 0 $: \[ f(0) = e^0 = 1 \] \[ f'(0) = e^0 = 1 \]

Step 3: Apply the Linear Approximation Formula

Using the linear approximation formula: \[ f(x) \approx f(a) + f'(a)(x - a) \] we substitute $ x = 0.02 $ and $ a = 0 $: \[ f(0.02) \approx f(0) + f'(0)(0.02 - 0) = 1 + 1 \cdot 0.02 = 1 + 0.02 = 1.02 \]