Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change

Note that we visited Equation of a Tangent Line here in the Definition of the Derivative section. Also, there are some Tangent Line Equation problems using the Chain Rule here in The Chain Rule section.

We will talk about the Equation of a Tangent Line with Implicit Differentiation here in the Implicit Differentiation and Related Rates section.

Equation of the Tangent Line

Let’s revisit the equation of a tangent line, which is a line that touches a curve at a point but doesn’t go through it near that point. The slope of the tangent line at this point of tangency, say “$a$”, is the instantaneous rate of change at $x=a$ (which we can get by taking the Derivative of the curve, as shown in the Basic Differentiation Rules section, and plugging in “$a$” for “$x$”). Since now we have the slope of this line, and also the coordinates of a point on the line, we can get the whole equation of this tangent line. We sometimes see this written as $\displaystyle \frac{{dy}}{{dx}}\left| {_{{x=a}}} \right.$.

When we get the derivative of a function, we’ll use the $x$-value of the point given to get the actual slope at that point. Then we’ll use the $y$-value of the point to get the complete line, using either the Point-Slope $(y-{{y}_{1}}=m\left( {x-{{x}_{1}}} \right))$ or Slope-Intercept $(y=mx+b)$ method. I like to use point-slope. It’s really not too bad! (Weird fact: the equation of a tangent line for a linear function is just that function!)

Here are some examples. Note that in the last problem, we are given a line parallel to the tangent line, so we need to work backwards to find the point of tangency, and then find the equation of the tangent line.

Here are other types of problems you might see:

Here is one more; in this problem, we’ll find the equation of a parabola that goes through a certain point and is tangent to a line at another point.

Equation of the Normal Line

The normal line to a curve at a point is the line through that point that is perpendicular to the tangent. Remember that a line is perpendicular to another line if their slopes are opposite reciprocals of each other; for example, if one slope is $4$, the other slope would be $\displaystyle -\frac{1}{4}$.

We do this problem the same way, but use the opposite reciprocal of the slope when we go to get the line; here is an example:

Horizontal and Vertical Tangent Lines

Sometimes we want to know at what point(s) a function has either a horizontal or vertical tangent line (if they exist). For a horizontal tangent line ($0$ slope), we want to get the derivative, and set it to $0$, which means setting the numerator to $\boldsymbol {0}$. Then we can get the $x$-value, and use the original function to get the $y$-value; thus, we have the point(s) at the horizontal tangent line. We also have to make sure that that the denominator isn’t $0$ at these points.

For a vertical tangent line (undefined slope), we want to get the derivative, and set the bottom or denominator to $\boldsymbol {0}$. Then we can get the $x$-value, and use the original function to get the $y$-value; thus, we have the point(s) at the vertical tangent line. We also to have to make sure the numerator isn’t $0$ at these points. Note that where functions have vertical tangent lines, they are not differentiable at that point.

Here are some examples:

Tangent Line Approximation

Also known as local linearization and linear approximation, it turns out that we can use a tangent line to approximate the value or graph of a function. The reason they want you to learn this stuff is so you can “appreciate the math” and how they used to use Calculus many years ago before fancy calculators and computers. And, actually, it’s pretty neat! (Note that we will show linear approximation using differentials here in the Differentials, Linear Approximation and Error Propagation section.)

Let’s say we have a function, such as  $f\left( x \right)=\sqrt{x}$, and we want to know what the value of this function is ($f\left( x \right)$), when $x=4.023$. We know it’s close to $2$, but we can get even closer without a calculator using Calculus. We can do this by using the point-slope formula $y-{{y}_{1}}=m\left( {x-{{x}_{1}}} \right)$, where $\left( {{{x}_{1}},\,\,{{y}_{1}}} \right)$ is a point that we can get easily, like $\left( {4,2} \right)$, and $m$ is the derivative of the function at that point. We can then get our new $y$-value, by plugging the original $x$-value we have ($4.03$).

Here is this problem, along with some others; use linear approximation (local linearization, or tangent line approximation) to find an approximation for the indicated values:

Rates of Change and Velocity/Acceleration

Note that there are more position, velocity, and acceleration problems here in the Antiderivatives and Indefinite Integration section. and in the Integration as Accumulated Change section.

The derivative has many applications in “real life”; one of the most useful is to find the rate of change of one variable with respect to another. Think of a rate of change, or sometimes called an instantaneous rate of change as how fast something is changing at a certain point, like a point in time. For example, if a particle is moving along a horizontal line, it’s position, relative to say the origin, is a function, but the derivative of this function is its instantaneous velocity (how fast it’s moving) at a certain point, and the derivative of its velocity is its acceleration (how fast its velocity is changing). Note that the acceleration function is a Higher Order Derivativesince we need to take the derivative of the position function twice to get the acceleration function. You can also think of the instantaneous velocity as the average velocity as the difference in $x$’s (for example, change in time) gets closer and closer to $0$; thus, we are taking a limit, or a derivative.

On the other hand, the average rate of change or average velocity is $\displaystyle \frac{{\text{change in position}}}{{\text{change in time}}}=\frac{{f\left( b \right)-f\left( a \right)}}{{b-a}}$ from interval $a$ to $b$, and you can think of this as $\displaystyle \text{rate}=\frac{{\text{distance}}}{{\text{time}}}$. These problems typically involve a time interval, as opposed to an instantaneous point. Note that we don’t need calculus to find this – just some algebra!

Again, typically when we are finding a rate of change, or instantaneous rate of change (“IROC”), we will be taking a derivative at a certain point, and when we are finding an average rate of change (“AROC”), we will be using the $\displaystyle \frac{{f\left( b \right)-f\left( a \right)}}{{b-a}}$ (slope) formula over an interval.

Here is a graph to show this; note how the instantaneous change is the slope of a tangent line at a point:
For these problems, find the average rate of change over the interval. Compare this to the instantaneous rate of change at the endpoints of the interval:

When we’re talking about an object traveling with respect to a time ($t$) with a position $s\left( t \right)$, velocity $v\left( t \right)$, and acceleration $a\left( t \right)$, here are some concepts. If you’ve taken Physics, you’ve probably dealt with this stuff:

• Initially is when $t=0$, at the origin is when $s(t)=0$, and at rest is when $v(t)=0$.
• If a particle is moving along a horizontal line, if its position $s(t)$ (relative to say the origin) is represented by a function, the derivative of this function is its instantaneous velocity $v(t)$ (how fast it’s moving), and the derivative of its velocity is its acceleration $a(t)$ (how fast its velocity is changing).
• The position of an object is a vector, since it has both a magnitude (a scalar, such as distance) and a direction. A change in position is the displacement vector, which is how far out of place the object is, compared to where it started. The distance it has traveled is the total amount of ground an object has covered during its motion, a scalar, and is the absolute value of the displacement vector.
• The velocity function is the derivative of the position function, and can be negative, zero, or positive. If the derivative (velocity) is positive, the object is moving to the right (or up, if that’s how the coordinate system is defined); if negative, it’s moving to the left (or down); if the velocity is 0, the object is at rest. Thus, if the velocity changes sign, the object changes direction.
• The velocity of an object is actually a vector, whereas the speed is the absolute value of the velocity vector, and is a scalar. The speed of an object cannot be negative, whereas velocity can.
• Total distance traveled is the sum of the absolute values of the differences in positions of all the resting points.
• Average Velocity is total displacement during that interval over total time elapsed.
• Acceleration (the derivative of velocity, which is also a vector) can cause speed to increase, decrease, or stay the same.
• If velocity and acceleration have the same sign, speed is increasing. (Example: a velocity from 40 mph to 45 mph, both positive, or from –40 mph to –45 mph, both negative).
• If velocity and acceleration have opposite signs, speed is decreasing. (Example: a velocity from 45 mph to 40 mph, velocity positive, acceleration negative, or from –45 mph to –40 mph, velocity negative, acceleration positive).
• Farthest left (or down) means minimum and farthest right (or up) means maximum.
• To find the maximum or minimum velocity or acceleration, find where the derivative of these functions fails or is zero.
• The position of free-falling objects is $\displaystyle s\left( t \right)=\frac{1}{2}g{{t}^{2}}+{{v}_{0}}t+{{s}_{0}}$, where ${{v}_{0}}$ is the initial velocity, ${{s}_{0}}$ is the initial height, and $g$ is the acceleration from gravity. On earth, the acceleration due to gravity is about –32 feet per second per second or –9.8 meters per second per second. In algebra, we saw this equation as $g\left( t \right)=-16{{t}^{2}}+{{v}_{0}}t+{{s}_{0}}$, since we typically used feet per second. When the object hits the ground, $s\left( t \right)=0$ (remember to always count the position from the ground up).

Here are some problems using the position, velocity, and acceleration functions:

Here’s one more problem, where we have to distinguish between the instantaneous velocity and the average velocity:

Learn these rules, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to The Chain Rule – you are ready!

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