Chapter 2, Problem 6PS

a.

To determine

To find the slope of the line joining two points.

Answer to Problem 6PS

  $\boxed{5}$

Explanation of Solution

Given information:

One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point $\left(2,4\right)$ on the graph of the quadratic function $f\left(x\right)=x^2$ as shown in the figure:

  

Find the slope of the line joining $\left(2,4\right)$ and $\left(3,9\right)$ .Is the slope of the tangent line $\left(2,4\right)$ at greater than or less than the slope of the line through $\left(2,4\right)$ and $\left(3,9\right)$ ?Calculation:

Here, we will consider a point $\left(2,4\right)$ on the graph of the quadratic equation $f\left(x\right)=x^2$ .

  

We will first differentiate the function $f\left(x\right)=x^2$ with respect to $x$ :

  $f\text{'}\left(x\right)=2x$

Now, the slope of the tangent line at point $\left(2,4\right)$ on the graph of $f\left(x\right)=x^2$ is:

  $f\text{'}\left(2\right)=2\left(2\right)$

  $=4$

As we know that, the slope of the line passing through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Hence, the slope of the line passing through the points $\left(2,4\right)$ and $\left(3,9\right)$ is given by:

  $m=\frac{9-4}{3-2}$ = $\boxed{5}$

Hence, the slope of the tangent line at point $\left(2,4\right)$ on the graph of $f\left(x\right)=x^2$ is less than the slope of the line through the points $\left(2,4\right)$ and $\left(3,9\right)$ .

b.

To determine

To find the slope $m_2$ of the line joining the two points.

Answer to Problem 6PS

  $\boxed{3}$

Explanation of Solution

Given information:

One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point $\left(2,4\right)$ on the graph of the quadratic function $f\left(x\right)=x^2$ as shown in the figure:

  

Find the slope $m_2$ of the line joining $\left(2,4\right)$ and $\left(1,1\right)$ . Is the slope of the tangent line at $\left(2,4\right)$ greater than or less than the slope of the line through $\left(2,4\right)$ and $\left(1,1\right)$ .

Calculation:

The slope of the line passing through the points $\left(2,4\right)$ and $\left(1,1\right)$ is given by:

  $m=\frac{4-1}{2-1}$ = $\boxed{3}$

Hence, the slope of the tangent line at point $\left(2,4\right)$ on the graph of $f\left(x\right)=x^2$ is greater than the slope of the line through the points $\left(2,4\right)$ and $\left(1,1\right)$ .

c.

To determine

To find the slope $m_3$ of the line joining the two points.

Answer to Problem 6PS

  $\boxed{4.1}$

Explanation of Solution

Given information:

One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point $\left(2,4\right)$ on the graph of the quadratic function $f\left(x\right)=x^2$ as shown in the figure:

  

Find the slope $m_3$ of the line joining $\left(2,4\right)$ and $\left(2.1,4.41\right)$ . Is the slope of the tangent line at $\left(2,4\right)$ greater than or less than the slope of the line through $\left(2,4\right)$ and $\left(2.1,4.41\right)$ .

Calculation:

The slope of the line passing through the points $\left(2,4\right)$ and $\left(2.1,4.41\right)$ is given by:

  $m=\frac{4.41-4}{2.1-2}=\frac{0.41}{0.1}=\boxed{4.1}$

Hence, the slope of the tangent line at point $\left(2,4\right)$ on the graph of $f\left(x\right)=x^2$ is less than the slope of the line through the points $\left(2,4\right)$ and $\left(2.1,4.41\right)$ .

d.

To determine

To find the slope $m_h$ of the line joining the two points.

Answer to Problem 6PS

  $\boxed{h+4}$

Explanation of Solution

Given information:

One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point $\left(2,4\right)$ on the graph of the quadratic function $f\left(x\right)=x^2$ as shown in the figure:

  

Find the slope $m_h$ of the line joining $\left(2,4\right)$ and $\left(2+h,f\left(2+h\right)\right)$ in terms of the nonzero number $h$ .

Calculation:

The slope of the line passing through the points $\left(2,4\right)$ and $\left(2+h,f\left(2+h\right)\right)$ for a nonzero number $h$ is given by:

  $\begin{array}{l}m=\frac{f\left(2+h\right)-4}{2+h-2}\\ m=\frac{{\left(2+h\right)}^2-4}{h}\\ m=\frac{4+h^2+4h-4}{h}\\ m=\frac{h^2+4h}{h}\\ m=\boxed{h+4}\end{array}$

Hence, the slope $m_h$ is $\boxed{h+4}$.

e.

To determine

To evaluate the slope formula.

Answer to Problem 6PS

For $h=-1$ , $\boxed{m=3}$

For $h=1$ , $\boxed{m=5}$

For $h=0.1$ , $\boxed{m=4.1}$

Explanation of Solution

Given information:

One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point $\left(2,4\right)$ on the graph of the quadratic function $f\left(x\right)=x^2$ as shown in the figure:

  

Evaluate the slope formula from part $\left(d\right)$ for $h=-1$ , $1$ and $0.1$ . Compare these values with those in parts $\left(a\right)-\left(c\right)$ .

Calculation:

For $h=-1$ , the slope will be:

  $\begin{array}{l}m=h+4\\ m=-1+4\\ m=\boxed{3}\end{array}$

For $h=1$ , the slope will be:

  $\begin{array}{l}m=h+4\\ m=1+4\\ m=\boxed{5}\end{array}$

For $h=0.1$ , the slope will be:

  $\begin{array}{l}m=h+4\\ m=0.1+4\\ m=4.1\end{array}$

Hence, the slope gives the same values as we get in parts $\left(a\right),\left(b\right)$ and $\left(c\right)$ .

f.

To determine

To write the conclusion about the slope $m_{\tan }$ .

Answer to Problem 6PS

The slope of the tangent line at point $\left(2,4\right)$ on the graph of $f\left(x\right)=x^2$ approaches to $4$ .

Explanation of Solution

Given information:

One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point $\left(2,4\right)$ on the graph of the quadratic function $f\left(x\right)=x^2$ as shown in the figure:

  

What can you conclude the slope $m_{\tan }$ of the tangent line at $\left(2,4\right)$ to be? Explain your answer.

Calculation:

Now, from the slope formula $m=h+4$ it can be concluded that as $h\to 0$ , the slope of the tangent line at a point $\left(2,4\right)$ on the graph of $f\left(x\right)=x^2$ approaches to $4$ .