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Section 1.10 Tangent Lines
Subsection 1.10.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 1.10.2 Introduction
In this activity, you will calculate tangent lines to functions, at various points and graph them along with the function. You will also find points on a function where the tangent line has a specific slope.
To find the tangent line to a function \(f(x)\) at \(x = a\text{,}\) two pieces of information are needed:
The point, \((a, f(a))\text{.}\)
The slope of the tangent line, \(f'(a)\text{.}\)
Aside A detailed example of finding and plotting tangent lines is described in
Example 12.1 .
Substituting these two pieces of information into the point-slope form of a line gives the following equation.
\begin{align}
y - y_0 \amp = m_{tan} \cdot (x-x_0) \nonumber\notag\\
y - f(a) \amp = f'(a) \cdot (x-a) \nonumber\notag\\
y \amp = f'(a) \cdot (x-a) + f(a)\tag{1.1}
\end{align}
You will need to use the resulting equation
(1.1) in the following exercises.
Exercises 1.10.3 Exercises
1. Assign the function
\(f(x) = \sqrt{9 - x}\) using the assignment operator
:=.
(a) Determine the equations of the tangent lines at
\(x=0\) and
\(x=5\text{.}\)
Hint .
Be sure to assign different names for each tangent line, such as
line1 and
line2. You will need to use unique names so that you can plot each line as well as
\(f(x)\) in the next step.
(b) Plot
\(f(x)\) and the two tangent lines using a single
plot() command.
Hint .
Since you are plotting more than one tangent line on the same axes, it is a good idea to specify plot colours. An example can be found in
Section 8.3 .
2. Assign the function
\(g(x)=x{ e}^{x}\text{.}\)
(a) Determine the equations of the tangent lines at
\(x=1\) and
\(x=-1\text{.}\)
Hint .
Make sure that you have properly defined your function using either
exp() as the exponential function or using the
\(e\) from the palettes toolbar.
(b) Plot
\(g(x)\) and the two tangent lines using a single
plot() command.
3. Assign the function
\(h(x)=x^3-x^2-9x+9\text{.}\) In this exercise, you will need to solve for values of
\(x\) where the tangent line has a specified slope.
(a) Determine the
\(x\) -values where the function has tangent lines with slope equal to
\(1\text{.}\)
(b) Find the
\(x\) -values where the function has horizontal tangent lines.
Hint .
You’ll need to remember what the slope of a horizontal tangent line is to solve for
\(x\text{.}\)
