Q. 2 - Calculus

Question

Examples: 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 Fundamental Theorem of Calculus does not apply on [0, 10].
(b) Three functions f that we do not yet know how to anti-differentiate.
(c) A non-constant function f for which ${\int }_0^{4}f\left(x\right). dx=0$.

Step-by-Step Solution

Answer

(a) A function f for which the Fundamental Theorem of Calculus does not apply on [0, 10] is ${\int }_0^{10}\frac{1}{x-2}.dx$

(b) Three functions f that we do not yet know how to anti differentiate are $f\left(x\right)=\tan x, g\left(x\right)={\tan }^{-1}x, h\left(x\right)=\tan hx$.

(c) The non-constant function is ${\int }_0^4(x-2)dx=0$

Part (a) Step 1. Given Information.

An interval $[0,10]$.

Part (a) Step 2. Find the function.

Consider the function $f\left(x\right)=\frac{1}{x-2}$.

The function is not continuous at $x=2$.

So the fundamental theorem does not apply on ${\int }_0^{10}\frac{1}{x-2}.dx$

Part (b) Step 1. Functions that does not have anti-differentiation.

The functions that does not have anti-differentiation are:

$$\begin{gathered} f\left(x\right)=\tan x, \\ g\left(x\right)={\tan }^{-1}x, \\ h\left(x\right)=\tan hx \end{gathered}$$

Part (c) Step 1. Consider the function.

Consider the non-constant function,

$f\left(x\right)=x-2$

$$\begin{gathered} {\int }_0^4(x-2)dx={[{\frac{x}{2}}^2-2x]}_0^4 \\ =[{\frac{4}{2}}^2-2(4\left)\right]-\left[0\right] \\ =\frac{16}{2}-8 \\ =0 \end{gathered}$$