Question

Find the instantaneous rate of change of the function f(x)=3x^2 as x approaches 3.

Answer 1

Answer:

The instantaneous rate of change as x approaches 3 is 18.

Step-by-step explanation:

From Differential Calculus and Geometry we remember that instantaneous rate of change of the function is represented by a tangent line, whose slope is determined by the first derivative of the curve. Let $f(x) = 3\cdot x^{2}$, the instantaneous rate of change of the function when x approaches 3 is deducted from the definition of derivative:

$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ (1)

$f'(x) = \lim_{h\to 0} \frac{3\cdot (x+h)^{2}-3\cdot x^{2}}{h}$

$f'(x) = \lim_{h\to 0} \frac{3\cdot (x^{2}+2\cdot x\cdot h +h^{2})-3\cdot x^{2}}{h}$

$f'(x) = \lim_{h\to 0} \frac{3\cdot x^{2}+6\cdot x\cdot h+3\cdot h^{2}-3\cdot x^{2}}{h}$

$f'(x) = \lim_{h\to 0} \frac{6\cdot x\cdot h +3\cdot h^{2}}{h}$

$f'(x) = \lim _{h\to 0} (6\cdot x+3\cdot h)$

$f'(x) = 6\cdot x \cdot \lim_{h\to 0} 1 + 3\cdot \lim_{h\to 0} h$

$f'(x) = 6\cdot x$ (2)

If we know that $x = 3$, then the instantaneous rate of change as x approaches 3 is:

$f'(3) = 6\cdot (3)$

$f'(3) = 18$

The instantaneous rate of change as x approaches 3 is 18.