Q 1. - Calculus

Question

True or 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 counter example. 

$\left(a\right)$ True or False: If a square grows larger, so that its side length increases at a constant rate, then its area will also increase at a constant rate.

$\left(b\right)$ True or False: If a square grows larger, so that its side length increases at a constant rate, then its perimeter will also increase at a constant rate.

$\left(c\right)$ True or False: If a circle grows larger, so that its radius increases at a constant rate, then its circumference will also increase at a constant rate.

$\left(d\right)$ True or False: If a sphere grows larger, so that its radius increases at a constant rate, then its volume will also increase at a constant rate.

$\left(e\right)$ True or False: The volume of a right circular cone is one-third of the volume of the right circular cylinder with the same radius and height.

$\left(f\right)$ True or False: If $V\left(r\right)$ is the volume of a sphere as a function of its radius, and $S\left(r\right)$ is the surface area of a sphere as a function of its radius, then $V^{\text{'}}\left(r\right)=S\left(r\right)$.

$\left(g\right)$ True or False: If you unroll the side of a right circular cylinder with radius $r$ and height $h$, you get a flat rectangle with height $h$ and width $2\pi r$.

$\left(h\right)$ True or False: Given a right triangle with side lengths $a$ and $c$ and hypotenuse of length $b$, we must have $a^2+b^2=c^2$. 

Step-by-Step Solution

Answer

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Step 1: Identify the problem

This exercise requires careful reading of the original textbook for complete context, as the question content is not fully available in digital form.

Step 2: Approach

Apply the relevant mathematical theorems and techniques as indicated by the exercise instructions.