Questions: Compute the values of dy and Δy for the function y=e^(2x)+2x given x=0 and Δx=dx=0.02

Compute the values of dy and Δy for the function y=e^(2x)+2x given x=0 and Δx=dx=0.02
Transcript text: Compute the values of $d y$ and $\Delta y$ for the function $y=e^{2 x}+2 x$ given $x=0$ and $\Delta x=d x=0.02$
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Solution

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Solution Steps

Step 1: Compute \(d y\)

To find \(d y\), we first find the derivative of the function, \(f'(x) = 2 e^{2 x} + 2\). Then, \(d y = f'(x_0) \cdot \delta = 4 \cdot 0.02 = 0.08\).

Step 2: Compute \(\Delta y\)

To find \(\Delta y\), we calculate the actual change in \(y\) by evaluating the function at \(x_0 + \delta\) and \(x_0\), i.e., \(\Delta y = f(x_0 + \delta) - f(x_0) = 1.081 - 1 = 0.0808\).

Final Answer:

The linear approximation of the change in \(y\), \(d y\), is 0.08. The actual change in \(y\), \(\Delta y\), is 0.0808.

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