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.)

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 $x$.

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

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 $a = 0$.

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

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

Final Answer