Q.1 - Calculus

Question

1. True/False: 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:$\int \left(5x^2-3x+2\right)dx$ 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 ${\int }_0^{2}f\left(x\right)g\left(x\right)dx=6$

(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

$2\sqrt{3}$

Step 1: Approach

Each true/false statement about calculus concepts requires careful examination of the precise mathematical definitions and potential counterexamples.

Step 2: General guidance

When evaluating such statements: if true, provide a proof or reference to a theorem; if false, give a specific counterexample that violates the claim.