Q. 2C - 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 $f$ whose area accumulation function is negative on $[0,5].$

(b) A function $f$ whose area accumulation function is decreasing on $[0,5].$

(c) Three antiderivatives of $e^{{\sin }^{2}x}.$

Step-by-Step Solution

Answer

Part (a) The example is $A\left(5\right)={\int }_0^5-t^{2}dt.$

Part (b) The example is $A\left(5\right)={\int }_0^5{t}^{2}dt.$

Part (c)  $F\left(x\right)={\int }_{-1}^x{e}^{{\sin }^{2}x}dt, F\left(x\right)={\int }_1^x{e}^{{\sin }^{2}x}dt, F\left(x\right)={\int }_2^x{e}^{{\sin }^{2}xdt}$

Part (a) Step 1. Calculation

Let us consider a function on $[0,5].$

$A\left(5\right)={\int }_0^5-t^{2}dt$

Value of the area is negative.

Therefore, the example is $A\left(5\right)={\int }_0^5-t^{2}dt.$

Part (b) Step 1. Calculation

Let us consider a function on $[0,5].$

$A\left(5\right)={\int }_0^5{t}^{2}dt$

The area accumulation function is decreasing on the interval.

Therefore, the example is $A\left(5\right)={\int }_0^5{t}^{2}dt.$

Part (c) Step 1. Calculation

The anti derivatives are,

$F\left(x\right)={\int }_{-1}^x{e}^{{\sin }^{2}x}dt, F\left(x\right)={\int }_1^x{e}^{{\sin }^{2}x}dt, F\left(x\right)={\int }_2^x{e}^{{\sin }^{2}xdt}$