# Introduction to Calculus

## Derivatives

Device and media usage changes at different rates for different groups of people. Communication and technology companies, marketers, educators, and their advocates maintain a close watch on trends and preferences. According to data from the Pew Research Center, Millennial ownership of smartphones only increased by one percent from 2018 to 2019 (from 92% to 93%). But for people over 74 years old, the number jumped from 30% to 40% in the same period.
Other device ownership and usage trends may go in different directions by generation. From 2018 to 2019, Millennial tablet computer ownership dropped from 64% to 52%. But during the same period, the Baby Boom generation's tablet computer ownership stayed exactly even with 52% reporting ownership. And the 74-and-older group's tablet ownership increased from 25% to 33%.https://www.pewresearch.org/fact-tank/2019/09/09/us-generations-technology-use/
What do these scenarios have in common? The functions representing them have changed over time. In this section, we will consider methods of computing such changes over time.

### Finding the Average Rate of Change of a Function

The functions describing the examples above involve a change over time. Change divided by time is one example of a rate. The rates of change in the previous examples are each different. In other words, some changed faster than others. If we were to graph the functions, we could compare the rates by determining the slopes of the graphs.
A tangent line to a curve is a line that intersects the curve at only a single point but does not cross it there. (The tangent line may intersect the curve at another point away from the point of interest.) If we zoom in on a curve at that point, the curve appears linear, and the slope of the curve at that point is close to the slope of the tangent line at that point.
 represents the function 
  We can see the slope at various points along the curve.


Let’s imagine a point on the curve of function 
  at 
  as shown in . The coordinates of the point are 
  Connect this point with a second point on the curve a little to the right of 
  with an x-value increased by some small real number 
  The coordinates of this second point are 
  for some positive-value 
  
We can calculate the slope of the line connecting the two points 
  and 
  called a secant line, by applying the slope formula,
 
  
We use the notation 
  to represent the slope of the secant line connecting two points.
The slope 
  equals the average rate of change between two points 
  and 
  

### Understanding the Instantaneous Rate of Change

Now that we can find the average rate of change, suppose we make 
  in  smaller and smaller. Then 
  will approach 
  as 
  gets smaller, getting closer and closer to 0. Likewise, the second point 
  will approach the first point, 
  As a consequence, the connecting line between the two points, called the secant line, will get closer and closer to being a tangent to the function at 
  and the slope of the secant line will get closer and closer to the slope of the tangent at 
  See .
Because we are looking for the slope of the tangent at 
  we can think of the measure of the slope of the curve of a function 
  at a given point as the rate of change at a particular instant. We call this slope the instantaneous rate of change, or the derivative of the function at 
  Both can be found by finding the limit of the slope of a line connecting the point at 
  with a second point infinitesimally close along the curve. For a function 
  both the instantaneous rate of change of the function and the derivative of the function at 
  are written as 
  and we can define them as a two-sided limit that has the same value whether approached from the left or the right.
The expression by which the limit is found is known as the difference quotient.

### Derivatives: Interpretations and Notation

The derivative of a function can be interpreted in different ways. It can be observed as the behavior of a graph of the function or calculated as a numerical rate of change of the function.
1. The derivative of a function 
  at a point 
  is the slope of the tangent line to the curve 
  at 
  The derivative of 
  at 
  is written 
  
2. The derivative 
  measures how the curve changes at the point 
  
3. The derivative 
  may be thought of as the instantaneous rate of change of the function 
  at 
  
4. If a function measures distance as a function of time, then the derivative measures the instantaneous velocity at time 
  

### Finding Derivatives of Rational Functions

To find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned.

### Finding Derivatives of Functions with Roots

To find derivatives of functions with roots, we use the methods we have learned to find limits of functions with roots, including multiplying by a conjugate.

### Finding Instantaneous Rates of Change

Many applications of the derivative involve determining the rate of change at a given instant of a function with the independent variable time—which is why the term instantaneous is used. Consider the height of a ball tossed upward with an initial velocity of 64 feet per second, given by 
  where 
  is measured in seconds and 
  is measured in feet. We know the path is that of a parabola. The derivative will tell us how the height is changing at any given point in time. The height of the ball is shown in  as a function of time. In physics, we call this the “s-t graph.”

### Using Graphs to Find Instantaneous Rates of Change

We can estimate an instantaneous rate of change at 
  by observing the slope of the curve of the function 
  at 
  We do this by drawing a line tangent to the function at 
  and finding its slope.

### Using Instantaneous Rates of Change to Solve Real-World Problems

Another way to interpret an instantaneous rate of change at 
  is to observe the function in a real-world context. The unit for the derivative of a function 
  is
Such a unit shows by how many units the output changes for each one-unit change of input. The instantaneous rate of change at a given instant shows the same thing: the units of change of output per one-unit change of input.
One example of an instantaneous rate of change is a marginal cost. For example, suppose the production cost for a company to produce 
  items is given by 
  in thousands of dollars. The derivative function tells us how the cost is changing for any value of 
  in the domain of the function. In other words, 
  is interpreted as a marginal cost, the additional cost in thousands of dollars of producing one more item when 
  items have been produced. For example, 
  is the approximate additional cost in thousands of dollars of producing the 12th item after 11 items have been produced. 
  means that when 11 items have been produced, producing the 12th item would increase the total cost by approximately $2,500.00.

### Finding Points Where a Function’s Derivative Does Not Exist

To understand where a function’s derivative does not exist, we need to recall what normally happens when a function 
  has a derivative at 
 . Suppose we use a graphing utility to zoom in on 
 . If the function 
  is differentiable, that is, if it is a function that can be differentiated, then the closer one zooms in, the more closely the graph approaches a straight line. This characteristic is called linearity.
Look at the graph in . The closer we zoom in on the point, the more linear the curve appears.
We might presume the same thing would happen with any continuous function, but that is not so. The function 
  for example, is continuous at 
  but not differentiable at 
  As we zoom in close to 0 in , the graph does not approach a straight line. No matter how close we zoom in, the graph maintains its sharp corner.
We zoom in closer by narrowing the range to produce  and continue to observe the same shape. This graph does not appear linear at 
  
What are the characteristics of a graph that is not differentiable at a point? Here are some examples in which function 
  is not differentiable at 
  
In , we see the graph of
Notice that, as 
  approaches 2 from the left, the left-hand limit may be observed to be 4, while as 
  approaches 2 from the right, the right-hand limit may be observed to be 6. We see that it has a discontinuity at 
  
In , we see the graph of 
  We see that the graph has a corner point at 
  
In , we see that the graph of 
  has a cusp at 
  A cusp has a unique feature. Moving away from the cusp, both the left-hand and right-hand limits approach either infinity or negative infinity. Notice the tangent lines as 
  approaches 0 from both the left and the right appear to get increasingly steeper, but one has a negative slope, the other has a positive slope.
In , we see that the graph of 
  has a vertical tangent at 
  Recall that vertical tangents are vertical lines, so where a vertical tangent exists, the slope of the line is undefined. This is why the derivative, which measures the slope, does not exist there.

### Finding an Equation of a Line Tangent to the Graph of a Function

The equation of a tangent line to a curve of the function 
  at 
  is derived from the point-slope form of a line, 
  The slope of the line is the slope of the curve at 
  and is therefore equal to 
  the derivative of 
  at 
  The coordinate pair of the point on the line at 
  is 
  
If we substitute into the point-slope form, we have
The equation of the tangent line is

### Finding the Instantaneous Speed of a Particle

If a function measures position versus time, the derivative measures displacement versus time, or the speed of the object. A change in speed or direction relative to a change in time is known as . The velocity at a given instant is known as instantaneous velocity.
In trying to find the speed or velocity of an object at a given instant, we seem to encounter a contradiction. We normally define speed as the distance traveled divided by the elapsed time. But in an instant, no distance is traveled, and no time elapses. How will we divide zero by zero? The use of a derivative solves this problem. A derivative allows us to say that even while the object’s velocity is constantly changing, it has a certain velocity at a given instant. That means that if the object traveled at that exact velocity for a unit of time, it would travel the specified distance.

### Key Equations


### Key Concepts

1. The slope of the secant line connecting two points is the average rate of change of the function between those points. See .
2. The derivative, or instantaneous rate of change, is a measure of the slope of the curve of a function at a given point, or the slope of the line tangent to the curve at that point. See , , and .
3. The difference quotient is the quotient in the formula for the instantaneous rate of change:
 
  
4. Instantaneous rates of change can be used to find solutions to many real-world problems. See .
5. The instantaneous rate of change can be found by observing the slope of a function at a point on a graph by drawing a line tangent to the function at that point. See .
6. Instantaneous rates of change can be interpreted to describe real-world situations. See  and .
7. Some functions are not differentiable at a point or points. See .
8. The point-slope form of a line can be used to find the equation of a line tangent to the curve of a function. See .
9. Velocity is a change in position relative to time. Instantaneous velocity describes the velocity of an object at a given instant. Average velocity describes the velocity maintained over an interval of time.
10. Using the derivative makes it possible to calculate instantaneous velocity even though there is no elapsed time. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, use the definition of derivative 
  to calculate the derivative of each function. 
For the following exercises, find the average rate of change between the two points.
For the following polynomial functions, find the derivatives.
For the following functions, find the equation of the tangent line to the curve at the given point 
  on the curve.
For the following exercise, find 
 
 such that the given line is tangent to the graph of the function.

### Graphical

For the following exercises, consider the graph of the function 
  and determine where the function is continuous/discontinuous and differentiable/not differentiable.
For the following exercises, use  to estimate either the function at a given value of 
  or the derivative at a given value of 
  as indicated.

### Technology


### Real-World Applications

For the following exercises, explain the notation in words. The volume 
  of a tank of gasoline, in gallons, 
  minutes after noon.
For the following exercises, explain the functions in words. The height, 
  of a projectile after 
  seconds is given by 
  
For the following exercises, the volume 
  of a sphere with respect to its radius 
  is given by 
  
For the following exercises, the revenue generated by selling 
  items is given by 
  
For the following exercises, the cost of producing 
  cellphones is described by the function 
  

### Extension

For the following exercises, use the definition for the derivative at a point 
  
  to find the derivative of the functions.

### Chapter Review Exercises


###  Finding Limits: A Numerical and Graphical Approach

For the following exercises, use .
For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as 
  approaches 
  If the function has limit as 
  approaches 
  state it.  If not, discuss why there is no limit.

### Finding Limits: Properties of Limits

For the following exercises, find the limits if 
  and 
  
For the following exercises, evaluate the limits using algebraic techniques.

### Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at 
  If not, describe the behavior of the graph of the function at 
  
For the following exercises, determine where the given function 
  is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

### 
  Derivatives
 

For the following exercises, find the average rate of change 
 

For the following exercises, find the derivative of the function.
For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

### Practice Test

For the following exercises, use the graph of 
  in .
For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as 
  approaches 
  If the function has a limit as 
  approaches 
  state it. If not, discuss why there is no limit
For the following exercises, evaluate each limit using algebraic techniques.
For the following exercises, determine whether or not the given function 
  is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.
For the following exercises, use the definition of a derivative to find the derivative of the given function at 
  
For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.
For the following exercises, explain the notation in words when the height of a projectile in feet, 
  is a function of time 
  in seconds after launch and is given by the function 
  
For the following exercises, use technology to evaluate the limit.
For the following exercises, consider the function whose graph appears in .
For the following exercises, use the function 
 .
For the following exercises, find the derivative of each of the functions using the definition: 
  