Linear Approximations and Applications

Question

Let 𝑦= 5𝑥^2 + 3𝑥 + 3. If Δ𝑥 = 0.3 at 𝑥 = 2, use linear approximation to estimate delta y.

Answer 1

delta y/ delta x = slope

use linear approx = > find y'

y'=10x+3

y'(2) = 23 => insta slope at x=2

y'(2) ~~delta y/ delta x = delta y / .3

=> delta y = y'(2) * delta x = 23 * .3 = 6.9

So, delta y is approximately 6.9 when delta x = .3 and x = 2.

Answer 2

Note: The following text is in LaTeX format. To view the equations more clearly, you can convert it to Word or PDF format using a tool like CloudConvert.

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

\[

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

\]

where \(f(x) = 5x^2 + 3x + 3\), and \(f'(x)\) is the derivative of \(f(x)\) with respect to \(x\).

First, let's find the derivative \(f'(x)\):

\[

f(x) = 5x^2 + 3x + 3

\]

\[

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

\]

\[

f'(x) = 10x + 3

\]

Next, evaluate \(f'(x)\) at \(x = 2\):

\[

f'(2) = 10(2) + 3 = 20 + 3 = 23

\]

Now, you need to use the linear approximation formula:

\[

\Delta y \approx f'(2) \cdot \Delta x = 23 \cdot 0.3 = 6.9

\]

So, the estimated value of \(\Delta y\) using linear approximation is \(6.9\).