Q2 - 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 graph of a function f on an interval [a, b] for which the left sum with four rectangles finds the exact area under the graph of f from x = a to x = b. (b) A graph of a function f on an interval [a, b] for which all trapezoid sums (regardless of the size of n) will find the exact area under the graph of f from x = a to x = b. (c) A graph of a function f on an interval [a, b] for which the upper sum with four rectangles is a much better approximation than the lower sum with four rectangles. 

Step-by-Step Solution

Answer

1. The ideal example of a function $f$ graph on interval [a, b] is  ${\int }_a^{b}5dx$, where the left sum with four rectangles gets the precise area under the graph from $\mathrm{x}=\mathrm{a}$ to $\mathrm{x}=\mathrm{b}$.
2. The required example for the graph of a function $f$ on interval $\left[a,b\right]$ for which all trapezoid sum finds the exact area under the graph from $\mathrm{x}=\mathrm{a}$ to $\mathrm{x}=\mathrm{b}$ is ${\int }_a^{b}5dx$.

3. The interval is,

   $[a, b]=[3,5]$

   The $n$value  is $4$.

Part (a) Step 1: Given information

The function is $f\left(x\right)=5$.

Part (a) Step2: Calculation

Consider that the function is;

$f\left(x\right)=5$

In this case, the left sum with four rectangles identifies the precise region under the graph from $x=a$ to $x=b$ . This is an example of a graph of a function on the interval $[a,b]$.

Therefore, the example is ${\int }_a^{b}5dx$.

.

Part (b) Step 1: Given information

The function is $f\left(x\right)=5$

Part (b) Step 2: Calculation

The function is,

$f\left(x\right)=5$

Finding an example of a graph of a function $f$ on an interval $[a, b]$ for which every trapezoid sum accurately determines the graph's area under the graph from$x=a to x=b$.

Therefore, the example is ${\int }_a^{b}5dx$.

Part (c) Step 1: Given information

The given function is $f=3x+1$

Part (c) Step 2: Calculation

Consider that the function is $f\left(x\right)=3 x+1$

The goal is to identify a situation when the higher sum for four rectangles approximates the bottom total for four rectangles significantly more accurately.

The function is,

$f\left(x\right)=3 x+1$

The interval is,

$[a, b]=[3,5]$

The required value of $n$ is $4$ .