Q. 2 - Calculus

Question

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. 

(a) A function f for which the signed area between f and the x-axis on [0, 4] is zero, and a different function g for which the absolute area between g and the x-axis on [0, 4] is zero. 

(b) A function f whose signed area on [0, 5] is less than its signed area on [0, 3]. 

(c) A function f whose average value on [−1, 6] is negative while its average rate of change on the same interval is positive. 

Step-by-Step Solution

Answer

(a). $f\left(x\right)=2x-4$

(b). $g\left(x\right)=4x-8$

(c). $f\left(x\right)=1$

(a). Given Information

$\text{ The objective is to construct an example for a function }f\text{ for which the signed area between }f$

$\text{ and the }x\text{-axis on }[0,4]\text{ is }0\text{ also another function }g\text{ for which the absolute area between }g$

$\text{ and the }x\text{-axis on }[0,4]\text{ is }0\text{. }$

$\text{ The example for signed area is, }$

$\begin{array}{r}{\int }_0^4 2x-4dx\\ ={\left[\frac{2x^2}{2}-4x\right]}_0^4\\ ={\left[x^2-4x\right]}_0^4\\ =16-16\\ =0\end{array}$

So,$f\left(x\right)=2x-4$ 

(b). Given Information

$\text{ The example for absolute area is, }$

$\begin{array}{r}\frac{1}{4-0}{\int }_0^4 4x-8dx\\ =\frac{1}{4}{\left[\frac{4x^2}{2}-8x\right]}_0^4\\ =\frac{1}{4}{\left[2x^2-8x\right]}_0^4\\ =\frac{1}{4}[32-32]\\ =0\end{array}$

 So, $g\left(x\right)=4x-8$

$\text{ Therefore, the examples are }f\left(x\right)=2x-4\text{ and }g\left(x\right)=4x-8$

(c). Given Information

$\text{ The objective is to construct an example for a function }f\text{ for which the signed area between }f$

$\text{ and the }x\text{-axis on }[0,5]\text{ is less than the signed area on }[0,3]\text{. }$

$\text{ The example is }f\left(x\right)=1$

$\text{ The signed area on }[0,5]\text{ is, }$

$\begin{array}{r}{\int }_0^5 1dx\\ =[x{]}_0^5\\ =5\end{array}$

$\text{ And, signed area on }[0,3]\text{ is }$

$\begin{array}{r}{\int }_0^3 1dx\\ =[x{]}_0^3\\ =3\end{array}$

$\text{ Therefore, the example is }f\left(x\right)=1\text{. }$