Question

For the equation y = 2 + 3 + 4x2 + 5x3
(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 #80250 – Math – Quantitative Methods

Question

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

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

Solution

$$L(x) = f(a) + f'(a)(x - a)$$ $$a = 3$$ $$f(3) = 2 + 9 + 36 + 135 = 182$$ $$f'(x) = 3 + 8x + 15x^2$$ $$f'(3) = 3 + 24 + 135 = 162$$ $$L(x) = 182 + 162 \cdot (x - 3)$$ $$L(x) = 182 + 162x - 486$$ $$L(x) = 162x - 304$$

Answer: $L(x) = 162x - 304$

Question

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

Solution

$$Q(x) = L(x) + \frac{f''(a)(x - a)^2}{2} = f(a) + f'(a)(x - a) + \frac{f''(a)(x - a)^2}{2}$$ $$f''(x) = 8 + 30x$$ $$f''(3) = 8 + 90 = 98$$ $$Q(x) = 162x - 304 + 49 \cdot (x - 3)^2$$ $$Q(x) = 162x - 304 + 49x^2 - 294x + 441$$ $$Q(x) = 49x^2 - 132x + 137$$

Answer: $Q(x) = 49x^2 - 132x + 137$

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