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RATES OF CHANGE & TANGENT LINES DAY 1AP Calculus AB
LEARNING TARGETS Define and determine the average rate of change Find the slope of a secant line Create the equation of the secant line Find the slope of a tangent line Explain the relationship of the slope of a secant line to
the slope of a tangent line Create the equation of the tangent line Define and determine the normal line Define and determine the instantaneous rate of change Solve motion problems using average/instantaneous rates of
change
AVERAGE RATE OF CHANGE In a population experiment, there were 150 flies on
day 23 and 340 flies on day 45. What was the average rate of change between day 23 and day 45?
In other words, what was the average rate of the flies growth between day 23 and day 45?
8.6 flies/day
AVERAGE RATE OF CHANGE:DEFINITION
The average rate of change of a function over an interval is the amount of change divided by the length of the interval.
In this class, we will often reference average rate of change as “AROC”. This is not necessarily a shortcut that is adapted across the board.
EXAMPLE Find the average rate of change of over the interval 1. 2.
12
One of the classic problems in calculus is the tangent line problem. You are probably very good at finding the slope of a line. Since the slope of a line (and line always implies straight in the world of math) is the same everywhere on the line, you could pick two points on the line and calculate the slope. With a curve, however, the slope is different depending where you are on the curve. You aren't able to just pick any old two points and calculate the slope. Think about the parabola,
2( ) 5 2f x x x
Whose graph looks like
Pretend you are a little bug and this graph is a mountain. The slope of the mountain is different at each point on the mountain. Think of how easy/hard it looks for the bug to climb. Just look at the bug’s angle.
Without calculus, we could estimate the slope at a particular point by choosing an additional point close to our point in question and then drawing a line between the points and finding the slope.
So if I was interested in finding out the slope at point P, I could estimate the slope by using a point Q, drawing a secant line (which crosses the graph in two places), and then finding the slope of that secant line.
P
SECANT LINE & SLOPE OF THE SECANT LINEDEFINITION
A line that joins two points on a function
The slope would be from the points (a,c) and (b,d)
SECANT LINE CONNECTION TO AROC & EXAMPLE Notice that the slope of the secant line is the average rate of change!
We could say that f(x) is the function which describes the growth of flies. A is at 23 days, b is at 45 days, c is 150, and d is 340.
We can see that the AROC is simple the slope of the secant line between the points (23, 150) and (45, 340)
HOW CAN WE FIND THE SLOPE OF TANGENT LINE?
Let’s take a look at a general secant line
What would be the slope of this secant line?
= How could we use this to
help us find the slope exactly at ?
Show applet
HOW CAN WE FIND THE SLOPE OF THE TANGENT LINE?
Let’s make the distance, , get smaller and smaller. Then, the slope would become more and more accurate.
The slope of the tangent line would be
TANGENT LINE & SLOPE OF THE TANGENT LINE DEFINITION
A straight line that touches the function at one point.
The slope of the tangent line would be when
SLOPE OF TANGENT EXAMPLE Find the slope of the tangent line of at the instant
1. Definition since a = 1 2.
= 3. Thus, the slope of the tangent line at is
INSTANTANEOUS RATE OF CHANGEDEFINITION
The rate of change at a particular instant/moment
In other words, it is the slope of the tangent line to the curve.
In this class, we will reference this as “IROC”
INSTANTANEOUS RATE OF CHANGEEXAMPLE
Let represent the daily growth of a plant in inches. What does the IROC represent at the instant .
1. Before, we saw that the slope at this point was 2.
2. Therefore, the IROC is that the plant is growing at a rate of 2 inches per day on the first day.
PRACTICE 1Find the average rate of change of the function over the interval . Then, write the equation of the secant line that describes that average rate of change.
1. Slope:
2. Use point slope form for equation:
3. m = 1, ,
4. or
PRACTICE 2Write the equation of the tangent for the function at the point
1.
2.
3. . Thus, m = 12.
4. The point is (2, 11)
5. Using point-slope form we get:
PRACTICE 3Determine whether the curve has a tangent at the indicated point. If it does, give its slope. If not, explain why not.
at the point x = 0
1. Is the function continuous? Yes (check all 3 conditions)
2. Check slope from both sides using one-sided limits
3.
4.
5. Limits do not match. Therefore, the slope does not exist at this point
PRACTICE 4Write the equation for the normal line to the curve at
(Normal line to a curve is the line perpendicular to the tangent at that point)
1. Normal is opp. recip. of tangent slope
2. . Thus,
3. Use point slope: