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

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Presentation on theme: "Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006."— Presentation transcript:

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 = ?


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?


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 ?


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!

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