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Miscellaneous Topics Calculus Drill!!. Developed by Susan Cantey at Walnut Hills H.S. 2006. Miscellaneous Topics. I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the AP Exam. When you think you know the answer,
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Miscellaneous TopicsCalculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006
Miscellaneous Topics • I’m going to ask you about various unrelated but important calculus topics. • It’s important to be fast as time is your enemy on the AP Exam. • 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.
1. What is the definition of LIMIT? OK…this is like the basis of ALL of Calculus. It was finally “perfected” by Cauchy in 1821. Ready?
Answer #1 Given any if there is a corresponding such that implies then we say that lim (This is the bare bones important part that you need to memorize…check your text for the detailed version.)
2. How many different methods are there for evaluating limits? Can you name several?
Inspection • Observe graph • Create a table of values • Re-write algebraically • Use L’Hopitals Rule • (only if the form is indeterminate) • 6. Squeeze theorem (rarely used!!)
1. 2. 3. 4. 5. 6. 7. Math Wars!!! Did you know all 7?
4. lim = ?
Hole – at x=3 in the example • Step – usually the function’s description is split up : • Vertical asymptote – at x=1 in the example f(x)={ for x<0 for x>0
7. Under what conditions does the derivative NOT exist at x=a
If there is a discontinuity at x=a or if there is a sharp corner, cusp or endpoint at x=a, then the derivative is undefined at x=a
A function that is either always increasing or always decreasing. (i.e. the derivative is always positive or always negative.)
11. Given (a,b) is on the graph of f(x)
Given f(x) h(x) g(x) near If both f(x) and g(x) as Then h(x) also.
13. What does the Intermediate Value Theorem say?
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.
14. What is the formula for the slope of the secant line through (a,f(a)) and (b,f(b)) and what does it represent?
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.
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.
16. What does = 0 tell you about the graph of ? Warning: irrelevant picture
The graph has a horizontal tangent line at x=a. f(a) might be a minimum or maximum…or perhaps there is just a horizontal inflection point.
17. 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?
FIRST DERIVATIVE TEST If changes from + to – at x=a then is a local maximum. If changes from – to + at x=a then is a local minimum. That’s a dam good test!
The Second Derivative Test: • Given then: • If , f(a) is a relative max • If , f(a) is a relative min • If the test fails Don’t be stumped! (lol)
20. What do you know about the graph of f(x) if (or does not exist)?
You know there might be an inflection point at x = a. (Check to see if there is also a sign change in at x = a to confirm the inflection point actually occurs)
Velocity = the first derivative of the position function, or v(b) = v(a) + (initial velocity + cumulative change in velocity)
acceleration = first derivative of velocity = second derivative of position
24. Using differentials to approximate with a point near on the tangent line… what does This is driving me nuts!
