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

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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.

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

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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!!)

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lim = ?

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What are the three main types of discontinuities?

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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)= {

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What is the definition of continuity at a point?

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What is a normal line?

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The line perpendicular to the tangent line.

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What does the Squeeze Theorem say?

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

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What does the Intermediate Value Theorem say?

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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.

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

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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.

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What does the Mean Value Theorem say?

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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.

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

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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.

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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?

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The derivative must change signs at x=a

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What is the First Derivative Test?

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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!!

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Whats the Second Derivative Test?

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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…

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

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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)

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How do you determine velocity?

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

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How do you determine speed?

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Speed = absolute value of velocity

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How do you determine acceleration?

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acceleration = first derivative of velocity = second derivative of position

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If f (x) is negative….

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Then f(x) is decreasing….

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If f (x) is positive….

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Then f(x) is increasing….

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If f (x) is negative then…

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f(x) is concave down

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If f (x) is positive then…

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f(x) is concave up

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How do you compute the average value of ?

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

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How do you locate and confirm vertical and horizontal asymptotes?

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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

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Back already?

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What is LHopitals Rule? ^

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

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What is the Fundamental Theorem of Calculus???

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

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

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

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

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

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Yea!!! Thats all folks!

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