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 3.2. 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.
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 3.5. 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 0. 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 :
These two expressions for calculating the slope of a secant line are illustrated in Figure 3.5. 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 3.6(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 3.6(b) shows that as the values of get closer to 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 3.6(c).
In Figure 3.8 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 In fact, the graph of itself appears to be locally linear in the immediate vicinity of
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.
First find the slope of the tangent line. In this example, use (3.1.1).
tan Apply the definition. Substitute and Factor the numerator to evaluate the limit.
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 3.11.
We can use (3.1.1), but as we have seen, the results are the same if we use (3.1.2).
tan Apply the definition. Substitute and Multiply numerator and denominator by to simplify fractions. Simplify. Simplify using for Evaluate the limit.
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 3.14.
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.
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 or if is given by the difference quotient
As the values of approach the values of ave 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 3.23. 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 (3.1.3) 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 (3.1.5).
The cost function for coffee is given by where is the number of thousand items made. Use a table to estimate the marginal cost of producing 1000 items and interpret.
Using (3.1.3), we can estimate the marginal cost at by computing a table of the slopes of the secant line using values of approaching . This is shown in Table 3.25.
Table3.25.Slopes of the secant lines using values of approaching 1
From the table we see that the slope of the secant line over the interval is the slope of the secant line over the interval is and so forth. Using this table of values, it appears that a good estimate is
When items are produced, the cost is decreasing at a rate of per thousand items made. So if we produce more items, we can expect costs to drop by .
A rock is dropped from a height of feet. Its height above ground at time seconds later is given by Find its instantaneous velocity second after it is dropped, using (3.1.3).
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.
Example3.28.Chapter Opener: Estimating Rate of Change of Velocity.
Figure3.29.(credit: modification of work by Codex41, Flickr)
Reaching a top speed of mph, the Hennessey Venom GT is one of the fastest cars in the world. In tests it went from to mph in seconds, from to mph in seconds, from to mph in seconds, and from to mph in seconds. Use this data to draw a conclusion about the rate of change of velocity (that is, its acceleration) as it approaches mph. Does the rate at which the car is accelerating appear to be increasing, decreasing, or constant?
A homeowner sets the thermostat so that the temperature in the house begins to drop from ° F at p.m., reaches a low of ° during the night, and rises back to ° by a.m. the next morning. Suppose that the temperature in the house is given by for where is the number of hours past p.m. Find the instantaneous rate of change of the temperature at midnight.
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 games are produced. Should the toy company increase or decrease production?
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.
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
ave
Instantaneous velocity is given by
We may estimate a derivative by using a table of values.
This book is a custom edition based on OpenStax Calculus Volume 1. You can download the original for free at https://openstax.org/details/books/calculus-volume-1.
