Question

1. (4 marks) use a table of values to estimate the instantaneous rate of change of $f(x)=x^2$ at $x=2$. use values of $x$ close to 2 from both sides.

Explanation:

Step1: Define average rate formula

The average rate of change of $f(x)$ between $x=a$ and $x=b$ is $\frac{f(b)-f(a)}{b-a}$. We use this for points near $x=2$.

Step2: Choose values left of 2

Select $x=1.9, 1.99, 1.999$ (approaching 2 from below):

• For $x=1.9$: $f(1.9)=(1.9)^2=3.61$

Rate: $\frac{f(1.9)-f(2)}{1.9-2}=\frac{3.61-4}{-0.1}=3.9$

• For $x=1.99$: $f(1.99)=(1.99)^2=3.9601$

Rate: $\frac{3.9601-4}{1.99-2}=\frac{-0.0399}{-0.01}=3.99$

• For $x=1.999$: $f(1.999)=(1.999)^2=3.996001$

Rate: $\frac{3.996001-4}{1.999-2}=\frac{-0.003999}{-0.001}=3.999$

Step3: Choose values right of 2

Select $x=2.1, 2.01, 2.001$ (approaching 2 from above):

• For $x=2.1$: $f(2.1)=(2.1)^2=4.41$

Rate: $\frac{4.41-4}{2.1-2}=\frac{0.41}{0.1}=4.1$

• For $x=2.01$: $f(2.01)=(2.01)^2=4.0401$

Rate: $\frac{4.0401-4}{2.01-2}=\frac{0.0401}{0.01}=4.01$

• For $x=2.001$: $f(2.001)=(2.001)^2=4.004001$

Rate: $\frac{4.004001-4}{2.001-2}=\frac{0.004001}{0.001}=4.001$

Step4: Estimate instantaneous rate

As $x$ approaches 2, the average rates approach 4.

Answer:

The instantaneous rate of change of $f(x)=x^2$ at $x=2$ is $\boldsymbol{4}$.