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Rate of change and tangent lines
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The average rate of change of a function over an interval is the amount of change divided by the length of the interval. On a graph this is equal to the slope of a secant line
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This graph shows the temperature of a cup of coffee over a 30 minute period. What is the average rate the coffee cools during the 1st 20 minutes?
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When the coffee was 1st made the temperature was
After 20 minutes the temperature was
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The line connecting these two points is a secant line.
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Find the slope of this line
Find the slope of this line. This will be the average change in temperature.
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Find the average rate of change of
Over the interval
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Find the slope of the secant
Find the average rate of change of Over the interval Find the slope of the secant
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Find the slope of the secant
The average rate of change from [-3,2] is 3.
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The slope of the tangent line gives the instantaneous rate of change.
Find the instantaneous rate of change of the temperature of the coffee at 5 min.?
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The slope of the tangent line at x=5 will give the instantaneous rate of change
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The slope of the tangent line will give the instantaneous rate of change
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Click Here to see an example of tangent lines
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Lets find the instantaneous rate of change at x=3
That is find the slope of the tangent line at x=3 We are going to write an expression for the slope of the secant from the point where x=3 to a second point. We will then look at what happens to the slope of the secant as the second point moves closer to (3,0). Demonstration
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Plug 3 into the equation Find the slope of the secant line between the point (3,0) and the generic point
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Find the slope of the secant line between the point (3,0) and the generic point
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You can graph this on your calculator to find the limit.
Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.
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You can graph this on your calculator to find the limit.
Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.
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You can graph this on your calculator to find the limit.
Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.
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You can graph this on your calculator to find the limit.
Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.
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You can graph this on your calculator to find the limit.
Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.
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You can graph this on your calculator to find the limit.
Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.
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You can graph this on your calculator to find the limit.
Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.
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You should have gotten -12
You should have gotten This is the instantaneous rate of change at x=3, the slope of the tangent at x=3 and the derivative at x=3
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Find the derivative at x = 3 using the limit of the slope of the secant.
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Let’s Review Find the average rate of change over the given interval.
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Find the derivative at x = -1 using the limit of the slope of the secant.
-(-1-3) 4
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A ball is thrown straight up from a rooftop 160 feet high with an initial velocity of 48 feet per second. The function Describes the balls height above the ground, s(t), in feet, t seconds after it is thrown. The ball misses the rooftop on its way down and eventually strikes the ground. What is the average velocity of the ball from 0 to 4 seconds? What is the instantaneous velocity at 3 seconds?
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Worksheet
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Find the equation of the tangent line at x=2
First find the derivative at x=2 which is the slope of the tangent line.
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Find the equation of the tangent line at x=2
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Find the equation of the tangent line at x=0
First find the derivative at x=0 which is the slope of the tangent line.
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Find the equation of the tangent line at x=0
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Wkst II
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