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Lesson 13 of 24
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# Introduction

The Hennessey Venom GT is one of the fastest cars in the world. In 2014, it reached a record-setting speed of 270.49 mph. It can go from 0 to 200 mph in 14.51 seconds. The techniques in this chapter can be used to calculate the acceleration the Venom achieves in this feat (see (Figure).)

Calculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science, and engineering, and it is critical to analysis in business and health as well. In this chapter, we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions in this chapter so that we can then explore applications of these techniques.

Now that we have both a conceptual understanding of a limit and the practical ability to compute limits, we have established the foundation for our study of calculus, the branch of mathematics in which we compute derivatives and integrals. Most mathematicians and historians agree that calculus was developed independently by the Englishman Isaac Newton (1643–1727) and the German Gottfried Leibniz (1646–1716), whose images appear in (Figure). When we credit Newton and Leibniz with developing calculus, we are really referring to the fact that Newton and Leibniz were the first to understand the relationship between the derivative and the integral. Both mathematicians benefited from the work of predecessors, such as Barrow, Fermat, and Cavalieri. The initial relationship between the two mathematicians appears to have been amicable; however, in later years a bitter controversy erupted over whose work took precedence. Although it seems likely that Newton did, indeed, arrive at the ideas behind calculus first, we are indebted to Leibniz for the notation that we commonly use today.

# Tangent Lines

We begin our study of calculus by revisiting the notion of secant lines and tangent lines. Recall that we used the slope of a secant line to a function at a point  to estimate the rate of change, or the rate at which one variable changes in relation to another variable. We can obtain the slope of the secant by choosing a value of  near  and drawing a line through the points  and , as shown in (Figure). The slope of this line is given by an equation in the form of a difference quotient:.

We can also calculate the slope of a secant line to a function at a value  by using this equation and replacing  with , where  is a value close to . We can then calculate the slope of the line through the points  and . In this case, we find the secant line has a slope given by the following difference quotient with increment :.

### Definition

Let  be a function defined on an interval  containing . If  is in , then

is a difference quotient.

Also, if  is chosen so that  is in , then

is a difference quotient with increment .

View several Java applets on the development of the derivative.

These two expressions for calculating the slope of a secant line are illustrated in (Figure). We will see that each of these two methods for finding the slope of a secant line is of value. Depending on the setting, we can choose one or the other. The primary consideration in our choice usually depends on ease of calculation.

In (Figure)(a) we see that, as the values of  approach , the slopes of the secant lines provide better estimates of the rate of change of the function at . Furthermore, the secant lines themselves approach the tangent line to the function at , which represents the limit of the secant lines. Similarly, (Figure)(b) shows that as the values of  get closer to 0, the secant lines also approach the tangent line. The slope of the tangent line at  is the rate of change of the function at , as shown in (Figure)(c).

You can use this site to explore graphs to see if they have a tangent line at a point.

In (Figure) we show the graph of  and its tangent line at  in a series of tighter intervals about . As the intervals become narrower, the graph of the function and its tangent line appear to coincide, making the values on the tangent line a good approximation to the values of the function for choices of  close to 1. In fact, the graph of  itself appears to be locally linear in the immediate vicinity of .

Formally we may define the tangent line to the graph of a function as follows.

### Definition

Let  be a function defined in an open interval containing . The tangent line to  at  is the line passing through the point  having slope

provided this limit exists.

Equivalently, we may define the tangent line to  at  to be the line passing through the point  having slope

provided this limit exists.

Just as we have used two different expressions to define the slope of a secant line, we use two different forms to define the slope of the tangent line. In this text we use both forms of the definition. As before, the choice of definition will depend on the setting. Now that we have formally defined a tangent line to a function at a point, we can use this definition to find equations of tangent lines.

### Finding a Tangent Line

Find the equation of the line tangent to the graph of  at .

#### Solution

First find the slope of the tangent line.

Next, find a point on the tangent line. Since the line is tangent to the graph of  at , it passes through the point . We have , so the tangent line passes through the point .

Using the point-slope equation of the line with the slope  and the point , we obtain the line . Simplifying, we have . The graph of  and its tangent line at  are shown in (Figure).

### The Slope of a Tangent Line Revisited

Use (Figure) to find the slope of the line tangent to the graph of  at .

#### Solution

The steps are very similar to (Figure).

We obtained the same value for the slope of the tangent line by using the other definition, demonstrating that the formulas can be interchanged.

### Finding the Equation of a Tangent Line

Find the equation of the line tangent to the graph of  at .

#### Solution

We can use (Figure), but as we have seen, the results are the same if we use (Figure).

We now know that the slope of the tangent line is . To find the equation of the tangent line, we also need a point on the line. We know that . Since the tangent line passes through the point  we can use the point-slope equation of a line to find the equation of the tangent line. Thus the tangent line has the equation . The graphs of  and  are shown in (Figure).

Find the slope of the line tangent to the graph of  at .

# The Derivative of a Function at a Point

The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the derivative. The process of finding a derivative is called differentiation.

### Definition

Let  be a function defined in an open interval containing . The derivative of the function  at , denoted by , is defined by

provided this limit exists.

Alternatively, we may also define the derivative of  at  as
provided this limit exists.

### Estimating a Derivative

For , use a table to estimate  using (Figure).

#### Solution

Create a table using values of  just below 3 and just above 3.

After examining the table, we see that a good estimate is .

For , use a table to estimate  using (Figure).

6

### Finding a Derivative

For , find  by using (Figure).

#### Solution

Substitute the given function and value directly into the equation.

### Revisiting the Derivative

For , find  by using (Figure).

#### Solution

Using this equation, we can substitute two values of the function into the equation, and we should get the same value as in (Figure).

For , find .

# Velocities and Rates of Change

Now that we can evaluate a derivative, we can use it in velocity applications. Recall that if  is the position of an object moving along a coordinate axis, the average velocity of the object over a time interval  if  if  is given by the difference quotient.

As the values of  approach , the values of  approach the value we call the instantaneous velocity at . That is, instantaneous velocity at , denoted , is given by.

To better understand the relationship between average velocity and instantaneous velocity, see (Figure). In this figure, the slope of the tangent line (shown in red) is the instantaneous velocity of the object at time  whose position at time  is given by the function . The slope of the secant line (shown in green) is the average velocity of the object over the time interval .

We can use (Figure) to calculate the instantaneous velocity, or we can estimate the velocity of a moving object by using a table of values. We can then confirm the estimate by using (Figure).

### Estimating Velocity

A lead weight on a spring is oscillating up and down. Its position at time  with respect to a fixed horizontal line is given by  ((Figure)). Use a table of values to estimate . Check the estimate by using (Figure).

#### Solution

We can estimate the instantaneous velocity at  by computing a table of average velocities using values of  approaching 0, as shown in (Figure).

From the table we see that the average velocity over the time interval  is 0.998334166, the average velocity over the time interval  is 0.9999833333, and so forth. Using this table of values, it appears that a good estimate is .

By using (Figure), we can see that.

Thus, in fact, .

A rock is dropped from a height of 64 feet. Its height above ground at time  seconds later is given by . Find its instantaneous velocity 1 second after it is dropped, using (Figure).

-32 ft/sec

#### Hint

. Follow the earlier examples of the derivative using (Figure).

As we have seen throughout this section, the slope of a tangent line to a function and instantaneous velocity are related concepts. Each is calculated by computing a derivative and each measures the instantaneous rate of change of a function, or the rate of change of a function at any point along the function.

### Definition

The instantaneous rate of change of a function  at a value  is its derivative .

### Chapter Opener: Estimating Rate of Change of Velocity

Reaching a top speed of 270.49 mph, the Hennessey Venom GT is one of the fastest cars in the world. In tests it went from 0 to 60 mph in 3.05 seconds, from 0 to 100 mph in 5.88 seconds, from 0 to 200 mph in 14.51 seconds, and from 0 to 229.9 mph in 19.96 seconds. Use this data to draw a conclusion about the rate of change of velocity (that is, its acceleration) as it approaches 229.9 mph. Does the rate at which the car is accelerating appear to be increasing, decreasing, or constant?

#### Solution

First observe that 60 mph = 88 ft/s, 100 mph  ft/s, 200 mph  ft/s, and 229.9 mph  ft/s. We can summarize the information in a table.

Now compute the average acceleration of the car in feet per second on intervals of the form  as  approaches 19.96, as shown in the following table.

The rate at which the car is accelerating is decreasing as its velocity approaches 229.9 mph (337.19 ft/s).

### Rate of Change of Temperature

A homeowner sets the thermostat so that the temperature in the house begins to drop from  at 9 p.m., reaches a low of  during the night, and rises back to  by 7 a.m. the next morning. Suppose that the temperature in the house is given by  for , where  is the number of hours past 9 p.m. Find the instantaneous rate of change of the temperature at midnight.

#### Solution

Since midnight is 3 hours past 9 p.m., we want to compute . Refer to (Figure).

The instantaneous rate of change of the temperature at midnight is  per hour.

### Rate of Change of Profit

A toy company can sell  electronic gaming systems at a price of  dollars per gaming system. The cost of manufacturing  systems is given by  dollars. Find the rate of change of profit when 10,000 games are produced. Should the toy company increase or decrease production?

#### Solution

The profit  earned by producing  gaming systems is , where  is the revenue obtained from the sale of  games. Since the company can sell  games at  per game,.

Consequently,.

Therefore, evaluating the rate of change of profit gives

Since the rate of change of profit A coffee shop determines that the daily profit on scones obtained by charging  dollars per scone is . The coffee shop currently charges  per scone. Find , the rate of change of profit when the price is  and decide whether or not the coffee shop should consider raising or lowering its prices on scones.

#### Solution

• The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment .
• The derivative of a function  at a value  is found using either of the definitions for the slope of the tangent line.
• Velocity is the rate of change of position. As such, the velocity  at time  is the derivative of the position  at time . Average velocity is given by.Instantaneous velocity is given by.
• We may estimate a derivative by using a table of values.
• # Key Equations

• Difference quotient
• Difference quotient with increment
• Slope of tangent line

• Derivative of  at

• Average velocity
• Instantaneous velocity