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 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

Part (a) The desired example of a graph of a function $f$ on interval $[a,b]$ for which the left sum with four rectangles finds the exact area under the graph from $x=a to x=b is {\int }_a^{b}5dx$.

Part (b) The desired example of a graph of a function $f$ on interval $[a,b]$ for which all trapezoid sum finds the exact area under the graph from $x=a to x=b is {\int }_a^{b}5dx$.

Part (c) The interval is, $[a,b]=[3,5]$.

Part (a) Step 1. Given information

The function is;

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

Part (a) Step 2. Calculation

Here we are finding example of a graph of a function $f$ on interval $[a,b]$ for which the left sum with four rectangles finds the exact area under the graph from $x=a to x=b$.

The function is,

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

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

Part (b) Step 1. Given information

The given function is,

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

Part (b) Step 2. Explanation

The objective is to find example of a graph of a function $f$ on interval $[a,b]$ for which all trapezoid sum finds the exact area under the graph from $x=a to x=b$.

The function is,

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

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

Part (c) Step 1. Given information

The function is,

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

Part (c) Step 2. Explanation

The objective is to find an example in which the upper sum for $4$ rectangles is a much better approximation than the lower sum with $4$ rectangles.

The function is,

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

The interval is,

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

The value of $n$ is $4.$