Question

For the equation y = 2 + 3 + 4x^2 + 5x^3
(a) Find the equation for the linear approximation when x = 3.
(b) Find the equation for the quadratic approximation, also when x = 3.

Answer 1

Answer on Question #80306 – Math – Quantitative Methods

Question

For the equation $y = 2 + 3 + 4x^2 + 5x^3$

(a) Find the equation for the linear approximation when $x = 3$.

(b) Find the equation for the quadratic approximation, also when $x = 3$.

Solution

Maybe the function should be

$$y = 2 + 3x + 4x^2 + 5x^3 \quad (1)$$

because it makes no sense to write $2+3$ instead of 5. I will do the question using the function (1).

We have

$$f(3) = 2 + 3 \cdot 3 + 4 \cdot 3^2 + 5 \cdot 3^3 = 182$$

Then

$$f'(x) = 3 + 8x + 15x^2,$$ $$f'(3) = 3 + 8 \cdot 3 + 15 \cdot 3^2 = 162,$$ $$f''(x) = 8 + 30x,$$ $$f''(3) = 8 + 30 \cdot 3 = 98.$$

(a) Linear approximation is

$$y = f(a) + f'(a)(x - a)$$

where $a = 3$.

Then we have

$$y = 182 + 162(x - 3)$$

or

$$y = -304 + 162x.$$

(b) Quadratic approximation is

$$y = f(a) + f'(a)(x - a) + \frac{1}{2}f''(a)(x - a)^2,$$

where $a = 3$.

Then we have

$$y = 182 + 162(x - 3) + \frac{98}{2}(x - 3)^2$$

or

$$y = 137 - 132x + 49x^2$$

Answer: a) $y = -304 + 162x$; b) $y = 137 - 132x + 49x^2$

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