Q. 1 C - Calculus

Question

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: The n-rectangle lower sum for $f$ on $[a,b]$ is always equal to the corresponding right sum.

(b) True or False: If $f$ is positive and increasing on $[a,b]$, then any left sum for $f$ on $[a,b]$ will be an under-approximation.

(c) True or False: If $f\left(a\right)>f\left(b\right)$, then any right-sum approximation for $f$ on $[a,b]$ will be an under-approximation.

(d) True or False: A midpoint sum is always a better approximation than a left sum.

(e) True or False: If $f$ is positive and concave up on all of $[a,b]$, then any left-sum approximation for $f$ on $[a,b]$ will be an under-approximation.

(f) True or False: If $f$ is positive and concave up on all of $[a,b]$, then any trapezoid sum approximation for $f$ on $[a,b]$ will be an over-approximation.

(g) True or False: An upper sum approximation for $f$ on $[a,b]$ can never be an under-approximation.  

(h) True or False: For every function $f$ on $[a,b]$, the left sum is always a better approximation with $10$ rectangles than with $5$ rectangles.

Step-by-Step Solution

Answer

Part (a). False

Part (b). True

Part (c). False

Part (d). False

Part (e). False

Part (f). True

Part (g). True

Part (h). False

Part (a) Step 1. Given information

The n -rectangle lower sum for $f$ on $[a,b]$ is always equal to the corresponding right sum. 

Part (b) Step 2. Calculation

The statement is,

The n- rectangular lower sum for $f$ on $[a,b]$ is always equal to the corresponding right sum.

The statement is false.

The upper sum is always greater than or equal to the actual signed area where as the lower sum is always less than or equal to the actual signed area.

Therefore, the statement is false

Part (b) Step 1. Given information

$[a,b]$If $f$ is positive and increasing on  then any left sum for $f$ on $[a,b]$ will be an under approximation. 

Part (b) Step 2. Calculation

The statement is true.

For an increasing function f, the left end points will be underestimate and hence the left sum is an under approximation.

Therefore, the statement is true

Part (c) Step 1. Given information

If $f\left(a\right)>f\left(b\right)$, then any right-sum approximation for $f$ on $[a,b]$ will be an under approximation.

Part (c) Step 2. Calculation

The statement is false.

For an increasing function $f$, the left end points will be under-approximation and hence the left sum is an under-approximation.

For an increasing function $f$, the right end points will be over-approximation and hence the left sum is an over-approximation.

Therefore, the statement is false.

Part (d) Step 1. Given information

A midpoint sum is always a better approximation than a left sum.

Part (d) Step 2. Calculation

The statement is false.

A midpoint sum is sometimes a better approximation than the left sum but not always.

Therefore, the statement is false

Part (e) Step 1. Given information

If $f$ is positive and concave up on all of $[a,b]$, then any left-sum approximation for $f$ on $[a,b]$ will be an under-approximation. 

Part (e) Step 2. Calculation

Since, $f$ is positive and concave up on all of $[a,b]$ so $f$ is a decreasing function.

For a decreasing function $f$, the left end points will be over-estimate and hence the left sum is an over- approximation.

Therefore, the statement is false.

Part (f) Step 1. Given information

If $f$ is positive and concave up on all of $[a,b]$, then any trapezoid sum approximation for $f$ on $[a,b]$ will be an over-approximation. 

Part (f) Step 2. Calculation

The trapezoid rule over-estimates a curve which is concave up.

Therefore, the statement is true.

Part (g) Step 1. Given information

An upper sum approximation for $f$ on $[a,b]$ can never be an under-approximation. 

Part (g) Step 2. Calculation

The upper sum is always greater than or equal to the actual signed area.

The upper sum is always over-approximation.

Therefore, the statement is true.

Part (h) Step 1. Given information

For every function $f$ on $[a,b]$, the left sum is always a better approximation with $10$ rectangles than with $5$ rectangles.

Part (h) Step 2. Calculation

The statement is,

An upper sum approximation for $f$ on $[a,b]$ can never be an under-approximation.

The upper sum is always greater than or equal to the actual signed area.

The upper sum is always over-approximation.

Therefore, the statement is true.