Q. 1 - Calculus

Question

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.

Part (a): The slope of the tangent line to a function f at the point $x = 4$ is given by $f\text{'}\left(4\right)$.

Part (b): The instantaneous rate of change of a function f at the point $x=-3$ is given by $f(-3)$.

Part (c): The instantaneous rate of change of a function f at a point $x=a$ can be represented as the slope of a secant line.

Part (d): Where a function f is positive, its associated slope function f ' is increasing.

Part (e): Where a function f is decreasing, its associated slope function f ' is negative.

Part (f): When a function f has a steep slope at a point on its graph, its instantaneous rate of change at that point will have a large magnitude.

Part (g): When the graph of a function f  is decreasing with a steep slope, the graph of the associated slope function f  is negative with a large magnitude.

Part (h): Suppose an object is moving in a straight path with position function $s\left(t\right)$. If data-custom-editor="chemistry" $\mathrm{s}\left(\mathrm{t}\right)$ is positive and decreasing, then the velocity $v\left(t\right)$ is negative.

Step-by-Step Solution

Answer

Part (a): The statement is true.

Part (b): The statement is true.

Part (c): The statement is true.

Part (d): The statement is true.

Part (e): The statement is true.

Part (f): The statement is true.

Part (g): The statement is true.

Part (h): The statement is true.

Part (a) Step 1. Determine if the statement is true or false.

The slope of the tangent line to a function at any point is given as $f\text{'}\left(x\right)$.

Therefore, at $x=4$, the slope of the function will be $f\text{'}\left(4\right)$.

Thus, the statement is true.

Part (b) Step 1. Determine if the statement is true or false.

The instantaneous rate of change of the function f at any point is given as $f\text{'}\left(x\right)$.

Therefore, at $x=-3$, the instantaneous rate is $f\text{'}\left(-3\right)$.

Thus, the statement is true.

Part (c) Step 1. Determine if the statement is true or false.

The instantaneous rate of change of the function f at any point $x=a$ can be represented as the slope of a secant line.

Thus, the statement is true.

Part (d) Step 1. Determine if the statement is true or false.

Consider two instantaneous point $$\begin{gathered} \mathrm{f}\left(1\right)=2, \\ \mathrm{f}\left(2\right)=0 \end{gathered}$$,

The instantaneous rate of change is given below,

$$\begin{gathered} =\frac{f\left(2\right)-f\left(1\right)}{2-1} \\ =\frac{0-2}{1} \\ =-2 \end{gathered}$$

Therefore, $f\text{'}\left(x\right)$ is decreasing.

Thus, the statement is false.

Part (e) Step 1. Determine if the statement is true or false.

Consider two instantaneous point $$\begin{gathered} \mathrm{f}\left(1\right)=2, \\ \mathrm{f}\left(2\right)=0 \end{gathered}$$,

The associate slope function has slope at $x=1$,

$$\begin{gathered} =\frac{f\left(2\right)-f\left(1\right)}{2-1} \\ =\frac{0-2}{1} \\ =-2 \end{gathered}$$

Therefore, $f\text{'}\left(x\right)$ is negative.

Thus, the statement is true.

Part (f) Step 1. Determine if the statement is true or false.

When the function has steep slope for any two instant the instantaneous rate of change at that point will be very high or it can be said it has larger magnitude.

Thus, the statement is true.

Part (g) Step 1. Determine if the statement is true or false.

When the function has steep slope for any two instant the instantaneous rate of change at that point will be very high or it can be said it has larger magnitude. Since f decreases with steep slope, the graph of associated slope function f' is negative with larger magnitude.

Thus, the statement is true.