1. Calculus I (MAT 145) Dr. Day Monday September 11, 2017
Return Quiz #5
Characteristics of a Continuous Function (2.5)
Secants Slopes to Tangent Slope: Using Limits (2.7)
Test #1: Friday!
Monday, September 11, 2017
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2. This implies three requirements:
A function f is continuous at x = a when the following limit statement can be verified at x = a:
This implies three requirements:
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3. Removable Discontinuity Removable Discontinuity
Infinite
Discontinuity
Jump
Discontinuity
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4. More about Continuity
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5. Identify intervals of continuity & Locations where function is continuous from right or left→ →
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6. Types of Discontinuities
Draw the graph of a function y = f(x) so that the graph shows:
exactly one removable discontinuity, and
exactly one jump discontinuity, and
exactly one infinite discontinuity.
After you have completed the graph, identify the location of each of these discontinuities.
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7. Monday, September 11, 2017
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8. Secant Slope
We want to find the tangent line to y = f (x) at the point P (a, f (a)).
Consider a nearby point Q (x, f (x)), where x  a, and compute the slope of the secant line PQ:
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9. Tangent Slope
Then we let Q approach P along the function y = f (x) by letting x approach a.
Secant slope is mPQ. Tangent line slope is m.
As Q approaches P, x approaches a.
And as x approaches a, mPQ approaches m.
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10. Tangent slope as limit
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11. Another way to label points and calculate slope
Let x = a + h and so the slope of the secant line PQ is
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12. Find an equation of the tangent line to the parabola y = x2 at the point P(1, 1).
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13. C B A D E
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14. Calculate the slope at x = −2. Calculate the slope at x = 0.
Calculate the slope at x = a.
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15. Calculate the slope of f(x) = x2 at x = a.
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16. We call this slope calculation the derivative of f at x = a.
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19. Monday, February 1, 2016
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20. Monday, September 11, 2017
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21. The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true derivative of f at x is the limit of the value of the slope function as the secant lines get closer and closer to being a tangent line.
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23. Monday, September 11, 2017
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24. The value f ’(a) is called:
the derivative of f at x = a,
the instantaneous rate of change of f at x = a,
the slope of f at x = a, and
the slope of the tangent line to f at x = a.
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25. The derivative in action!
S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100?
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26. The derivative in action!
S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100?
From the description of the context, the “rate units” are: feet per minute.
The value 12 is an input variable, so we are looking at the precise instant that 12 minutes of travel has occurred, since some designated starting time when t = 0.
S’ indicates rate of change of S, indicating we have information about how S is changing with respect to t, in feet per minute.
The value 100 specifies the rate: 100 feet per minute.
Putting it all together:
At precisely 12 minutes into the trip, the object’s position is increasing at the rate of 100 feet per minute.
Monday, September 11, 2017
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27. The derivative in action!
C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5?
Monday, September 11, 2017
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28. The derivative in action!
C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5?
From the description of the context, the “rate units” are: thousands of dollars per patient.
The value 90 is an input variable, so we are looking at the precise instant when 90 patients are in the hospital.
C’ indicates rate of change of C, indicating we have information about how C is changing with respect to p, in thousands of dollars per patient.
The value 4.5 specifies the rate: 4.5 thousand dollars ($4500) per patient.
Putting it all together:
At precisely the instant that 90 patients are in the hospital, the cost per patient is increasing at the rate of $4500 per patient.
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29. The derivative in action!
V(r) represents the volume of a sphere, where r is the radius of the sphere in cm. What is the meaning of V ’(3)=36π?
From the description of the context, the “rate units” are: cubic cm of volume per cm of radius.
The value 3 is an input variable, so we are looking at the precise instant when the sphere’s radius is 3 cm long.
V’ indicates rate of change of V, indicating we have information about how V is changing with respect to r, in cubic cm per cm.
The value 36π specifies the rate: 36π cubic cm of volume per 1 cm of radius length.
Putting it all together:
At precisely the instant that the sphere has a radius length of 3 cm, the sphere’s volume is increasing at the rate of 36π cubic cm per cm of radius length.
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30. Tangent slope again
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31. Find an equation of the tangent line to the parabola y = x2 at the point P(1, 1).
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32. Can we create a derivative function f that will be true for any x value where a derivative exists?
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33. Calculate the derivative function, f ’(x), for f(x) = x2
Calculate the derivative function, f ’(x), for f(x) = x2. Use the limit definition of the derivative.
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34. Monday, September 11, 2017
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35. Here is a graph of the function y = g(x)
Here is a graph of the function y = g(x). Arrange the following values in increasing order. Explain your process and determination.
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36. Here is the graph of the function y = |x|.
Why does the derivative NOT exist at x = 0?
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37. Three situations for which a derivative DOES NOT EXIST!
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38. For each graphed function, state points at which the function is NOT differentiable. Explain your choices!
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39. Match each function, a-d, with its derivative, I-IV.
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40. Identify each curve. Explain your choices.
Here are the graphs of four functions. One repre- sents the position of a car as it travels, another represents the velocity of that car, a third repre- sents the acceleration of the car, and a fourth graph represents the jerk for that car.
Identify each curve. Explain your choices.
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41. Here is the graph of a function f. Use it to sketch the graph of f ’.
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