Linear Approximations and Applications

Question

Let y = 4x^2 +5x + 3. If delta x = 0.2 and x = 2 show how to use linear approximation to estimate delta y.

delta y is approximately? _______

Please show steps with explanation. Not a video.

Answer 1

Δy≈f'(x)Δx

Δx=0.2 as given

f(x) = 4x²+5x+2

f'(x)=8x+5

f'(2) = (8(2)+5) = 21

Δy ≈ f '(2)(0.2) = 21(.2) = 4.2

Check against actual Δy

Δy = f(x+Δx) - f(x) = f(2.2) - f(2) = 32.36 - 28 = 4.36

Answer 2

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To estimate \(\Delta y\) using linear approximation, you need to use the formula:

\[

\Delta y \approx f'(x) \cdot \Delta x

\]

where \(f'(x)\) is the derivative of the function at \(x = 2\) and \(\Delta x = 0.2\).

Given \(y = 4x^2 + 5x + 3\), let's find the derivative \(f'(x)\):

\[

f'(x) = \frac{d}{dx}(4x^2 + 5x + 3) = 8x + 5

\]

Next, you need to evaluate the derivative at \(x = 2\):

\[

f'(2) = 8(2) + 5 = 16 + 5 = 21

\]

Now, you need to use the linear approximation formula:

\[

\Delta y \approx 21 \cdot 0.2 = 4.2

\]

So, \(\Delta y\) is approximately 4.2.