Question

Find the instantaneous rate of change of the function f(x) = x2 + 3x + 4 when x = 2 using First Principles. Confirm that your answer is correct using the derivative rules.

Answer 1

$f(x)=x^2+3x+4$

$a=2$

Formula for instantaneous rate of change is  $lim_{h\rightarrow0}\dfrac{f(a+h)-f(a)}{h}$

$lim_{h\rightarrow0}\dfrac{f(2+h)-f(2)}{h}$

$lim_{h\rightarrow0}\dfrac{[(2+h)^2+3(2+h)+4]-14}{h}$

$lim_{h\rightarrow0}\dfrac{h(h+7)}{h}$

$lim_{h\rightarrow0}(h+7)= 7$

Derivative rule:

$\dfrac{d}{dx}(x^2+3x+4)=2x+3$

at x=2

$f(x)= 2(2)+3=7$

Hence, the answer is confirmed