3-12-12 - 2.1 tangent lines and rates of change...

31212 2.1 Tangent Lines and Rates of Change Notes.notebook

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PAGE 131

CHAPTER 2 THE DERIVATIVEMany realworld phenomena involve changing quantities the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a "derivative," which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation.

One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant.

2.1 Tangent Lines and Rates of ChangeTo understand what the derivative tells you, let's look at 2 types of lines: a secant line and a tangent line

secant line touches graph at 2 points

tangent line touches graph at exactly 1 point

If we let x + h approach x, then the point Q will move along the curve and approach the point P. If the secant line through P and Q approaches a limiting position as x + h ⇒ x, then we will regard that position to be the position of the tangent line at P. Stated another way, if the slope mPQ of the secant line through P and Q approaches a limit as x + h ⇒ x, then we regard the limit to be the slope mtan of the tangent line at P. Thus, we make the following definition:

Suppose that x0 is in the domain of the function f. The tangent line to the curve y = f(x) at the point P(x0, f(x0)) is the line with the equation

y f(x0) = mtan(x x0)

where

provided the limit exists. For simplicity, we will also call this the tangent line to y = f(x) at x0.

http://www.csulb.edu/~wziemer/TangentLine/TangentLine.html

http://www.csulb.edu/~wziemer/tangentline/tangentline.html


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EXAMPLES1) Find an equation for the tangent line to the curve y = 2x2 + 1 at the point where x 0 = 1 on this curve.

2) Find an equation of the tangent line to the curve y = x at the point where x0 = 4 on this curve.

VelocityOne of the important themes in calculus is the study of motion. To describe the motion of an object completely, one must specify its speed (how fast it is going) and the direction in which it is moving. The speed and direction of motion together comprise what is called the velocity of the object. For example, knowing that the speed of an aircraft is 500 mi/h tells us how fast it is going, but not which way it is moving. In contrast, knowing that the velocity of the aircraft is 500 mi/h due south pins down the speed and the direction of motion.

Later, we will study the motion of objects that move along curves in two or threedimensional space, but for now we will only consider motion along a line; this is called rectilinear motion. Some examples are a piston moving up and down in a cylinder, a race car moving along a straight track, an object being dropped from the top of a building and falling straight down, a ball thrown straight up and then falling down along the same line, and so forth.

For computational purposes, we will assume that a particle in rectilinear motions moves along a coordinate line, which we will call the saxis. A graphical description of rectilinear motion along an saxis can be obtained by making a plot of the scoordinate of the particle versus the elapsed time t from starting time t = 0. This is called the position versus time curve for the particle.

If a particle in rectilinear motion moves along an saxis so that its position coordinate function of the elapsed time t is

s = f(t)

then f is called the position function of the particle. The average velocity of a particle over a time interval is:

st

s

change in position time elapsed

http://www.calculusapplets.com/avevel.html

http://www.calculusapplets.com/avevel.html


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March 27, 2014EXAMPLESuppose that s = f(t) = 2 + 7t 2t2 is the position function of a particle, where s is in meters and t is in seconds. Find the average velocity of the particle over the time interval from [0, 3].

For a particle in rectilinear motion, average velocity describes its behavior over an interval of time. Sometimes we are interested in the particle's "instantaneous velocity" which describes its behavior at a specific instant in time.

EXAMPLEConsider the particle in the previous example, whose function is

s = f(t) = 2 + 7t 2t2

The position of the particle at time t = 2 sec is s = 8 m. Find the particle's instantaneous velocity at time t = 2 s.

Slopes and Rates of ChangeVelocity can be viewed as rate of change the rate of change of position with respect to time refers to a 'rate of change in y with respect to x.' See examples on page 137.

So if we were to find the rate of change of y with respect to x for the following function: y = 3x 2 it would be

The following formulas can also be used when trying to find a specific rate of change.If y = f(x) then we define:

1) the average rate of change of y with respect to x over the interval [x0, x1] to be

2) the instantaneous rate of change of y with respect to x at x0 to be

or

or

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EXAMPLELet y = x2 + x 1

a) Find the average rate of change of y with respect to x over the interval [2, 7].

b) Find the instantaneous rate of change of y with respect to x when x = 2.

Rates of Change in ApplicationsIn applied problems, average and instantaneous rates of change must be accompanied by appropriate units. In general, the units for a rate of change of y with respect to x are obtained by 'dividing' the units of y by the units of x and then simplifying accordingly to the standard rules of algebra. Look at the examples on page 139.

HW: PAGES 141 142, PROBLEMS 3, 4, 8, 9, 13, 16, 18, 23, 27, 28


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secant line touches graph at 2 points

tangent line touches graph at exactly 1 point

3-12-12 - 2.1 tangent lines and rates of change...

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