Download presentation

Presentation is loading. Please wait.

Published byKatherine Hahn Modified over 2 years ago

1
Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006

2
Miscellaneous Topics Im going to ask you about various unrelated but important calculus topics. When you think you know the answer, (or if you give up ) click to get to the next slide to see if you were correct.

3
How many different methods are there for evaluating limits? Can you name several?

4
1. Inspection 2. Observe graph 3. Create a table of values 4. Re-write algebraically 5. Use LHopitals Rule (only if the form is indeterminate) 6. Squeeze theorem (rarely used!!)

5
lim = ?

6

7
What are the three main types of discontinuities?

8
1. Hole – at x=3 in the example 2. Step – usually the functions description is split up : 3. Vertical asymptote – at x=1 in the example for x<0 for x>0 f(x)= {

9
What is the definition of continuity at a point?

10

11
What is a normal line?

12
The line perpendicular to the tangent line.

13
What does the Squeeze Theorem say?

14
If both f(x) and g(x) as Then h(x) also. Given f(x) > h(x) > g(x) near

15
What does the Intermediate Value Theorem say?

16
If f(x) is continuous and p is a y-value between f(a) and f(b), then there is at least one x-value between a and b such that f(c) = p.

17
What is the formula for the slope of the secant line through (a,f(a)) and (b,f(b)) and what does it represent?

18
average rate of change in f(x) from x=a to x=b Note: This differs from the derivative which gives exact instantaneous rate of change values at single x-value but you can use it to the derivative value at some values of x=c between a and b.

19
What does the Mean Value Theorem say?

20
If f(x) is continuous and differentiable, then for some c between a and b That is the exact rate of change equals the average (mean) rate of change at some point in between a and b.

21
What does f (a) = 0 tell you about the graph of f(x) ? Warning: irrelevant picture

22
The graph has a horizontal tangent line at x=a. f(a) might be a minimum or maximum…or perhaps just a horizontal inflection point.

23
What else must happen in addition to the derivative being zero or undefined at x=a in order for f(a) to be an extrema?

24
The derivative must change signs at x=a

25
What is the First Derivative Test?

26
FIRST DERIVATIVE TEST If f (x) changes from + to – at x=a then f(a) is a local maximum. If f (x) changes from – to + at x=a then f(a) is a local minimum. Dam thats a good test!! Dam, thats a great test!!

27
Whats the Second Derivative Test?

28
Given f (a)=0 then: 1.If f (a) < 0, f(a) is a relative max 2.If f (a) > 0, f(a) is a relative min 3.If f (a) = 0 the test fails The Second Derivative Test: Dont be Stumped... Ha ha ha…

29
What do you know about the graph of f(x) if f (a) = 0 (or does not exist)?

30
You know there might be an inflection point at x = a. (Check to see if there is also a sign change in f at x = a to confirm the inflection point actually occurs)

31
How do you determine velocity?

32
Velocity = the first derivative of the position function, or v(a) + (initial velocity + cumulative change in velocity)

33
How do you determine speed?

34
Speed = absolute value of velocity

35
How do you determine acceleration?

36
acceleration = first derivative of velocity = second derivative of position

37
If f (x) is negative….

38
Then f(x) is decreasing….

39
If f (x) is positive….

40
Then f(x) is increasing….

41
If f (x) is negative then…

42
f(x) is concave down

43
If f (x) is positive then…

44
f(x) is concave up

45
How do you compute the average value of ?

46
______________________ b - a dx Note: This is also known as the Mean (average) Value Theorem for Integrals

47
How do you locate and confirm vertical and horizontal asymptotes?

48
Vertical – suspect them at x-values which cause the denominator of f(x) to be zero. Confirm that the limit as x a is infinite…. Horizontal – suspect rational functions Confirm that as x, y a

49
Back already?

50
What is LHopitals Rule? ^

51
Given that as x both f and g or both f and g then the limit of = the limit of as x LHopitals Rule: ^

52
What is the Fundamental Theorem of Calculus???

53
where F (x) = f(x) Do you know the other form? The one that is less commonly used? The FUN damental Theorem of Calculus:

54
What is the general integral for computing volume by slicing (disk method)? (Assume we are revolving f(x) about the x-axis)

55
What if we revolve f(x) around y=a ?

56

57
What if we revolve the area between 2 functions: f(x) and g(x) around the x-axis?

58
Be sure to square the radii separately!!! (and put the larger function first)

59
Yea!!! Thats all folks!

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google