"How do you find the instantaneous rate of..." was answered by students like you

Kaiya Hardin

Answered question

2022-04-21

How do you find the instantaneous rate of change of

y = 4 x^{3} + 2 x - 3

at x=2?

Elliot Roy

Beginner2022-04-22Added 14 answers

The instantanous rate of change of the function f(x) in x=a is the slope of the tangent line to the graph of the function in that point:
m=y'(a).
So:

y^{'} = 12 x^{2} + 2

and:

y^{'} (2) = 12 \cdot 4 + 2 = 50

.
The answer is: 50.

Marely Anthony

Beginner2022-04-23Added 8 answers

Use the definition:

lim_{h \to 0} \frac{f (2 + h) - f (2)}{h}

lim_{h \to 0} \frac{f (2 + h) - f (2)}{h}

= lim_{h \to 0} \frac{[4 {(2 + h)}^{3} + 2 (2 + h) - 3] - [4 {(2)}^{3} + 2 (2) - 3]}{h}

= lim_{h \to 0} \frac{[4 (8 + 12 h + 6 h^{2} + h^{3}) + 2 (2 + h) - 3] - [33]}{h}

= lim_{h \to 0} \frac{[32 + 48 h + 24 h^{2} + 4 h^{3} + 4 + 2 h - 3] - [33]}{h}

= lim_{h \to 0} \frac{50 h + 24 h^{2} + 4 h^{3}}{h}

= lim_{h \to 0} (5) + 24 h + 4 h^{2}) = 50

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