Q. 1C - 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 left-sum and right-sum approximations are the same if the number n of rectangles is very large.

(b) True or False:${\int }_{-2}^5{(x+2)}^{3}dx$ is a real number.

(c) True or False: $\int (5x^2-3x+2)dx$ is exactly $26.167$

(d) True or False: ${\int }_{-3}^{-2}f\left(x\right)dx=-{\int }_2^{3}f\left(x\right)dx$

(e) True or False: If ${\int }_0^{2}f\left(x\right)dx=3 and {\int }_0^{2}g\left(x\right)dx=2 then {\int }_0^{2}f\left(g\right(x\left)\right)=6.$

(f) True or False: If ${\int }_0^{2}f\left(x\right)dx=3 and {\int }_0^{2}g\left(x\right)dx=2 then {\int }_0^{2}f\left(x\right)g\left(x\right)=6.$

(g) True or False: If ${\int }_0^{1}f\left(x\right)dx=3 and {\int }_{-1}^{0}f\left(x\right)dx=4, then {\int }_{-1}^{1}f\left(x\right)dx=7.$

(h) True or False: If ${\int }_0^{2}f\left(x\right)dx=3 and {\int }_2^{4}g\left(x\right)dx=4, then {\int }_0^4\left(f\right(x)+g(x\left)\right)dx=7.$

Step-by-Step Solution

Answer

Part (a) The given statement is false.

Part (b) The given statement is true.

Part (c) The given statement is false.

Part (d) The given statement is false.

Part (e) The given statement is false.

Part (f) The given statement is true.

Part (g) The given statement is true.

Part (h) The given statement is false.

Part (a) Step 1. Explanation

The left sum and the right sum approximations are the same if the number $n$ of rectangles is very large

The statement is false.

Part (b) Step 1. Explanation

The approximation of right sum is always greater than the left sum.

Therefore, the statement is false

The exact value of the definite integral ${\int }_{-2}^5{(x+2)}^{3}dx$ is a real number.

Since all the parameters are real, its exact value is also real.

Therefore, the statement is true.

Part (c) Step 1. Explanation

The exact value of $\int (5x^2-3x+2)dx$ is exactly $26.167$

The statement is false.

Since the upper and the lower limits of the integral is not known hence its exact value could not be found.

Therefore, the statement is false.

Part (d) Step 1. Explanation

The equation is false.

since, ${\int }_{-3}^{-2}f\left(x\right)dx=-{\int }_{-2}^{-3}f\left(x\right)dx$

Therefore, the statement is false.

Part (e) Step 1. Explanation

The statement is false since there is not enough information for which ${\int }_0^{2}f\left(g\right(x\left)\right)dx=6$ could be computed

Therefore, the statement is false.

Part (f) Step 1. Explanation

The statement is false since there is not enough information for which ${\int }_0^{2}f\left(x\right)g\left(x\right)dx=6$ could be computed.

Therefore, the statement is false.

Part (g) Step 1. Explanation

The statement is true.

Since,

$$\begin{gathered} {\int }_{-1}^{1}f\left(x\right)dx \\ ={\int }_{-1}^{0}f\left(x\right)dx+{\int }_0^{1}f\left(x\right)dx \\ =4+3 \\ =7 \end{gathered}$$

Therefore, the statement is true.

Part (h) Step 1. Explanation

The statement is false.

Since, the limits of both the expressions are not the same hence their addition is not possible.

Therefore, the statement is false.