Miscellaneous Topics Calculus Drill!! Miscellaneous Topics Im going to ask you about various unrelated but important calculus topics. Its 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. How many different methods are there for evaluating limits? Can you name several? 1. Direct Substitution 2. Observe graph 3. Create a table of values 4. Re-write algebraically (Simplify) 5. Use LHopitals Rule (only if the form is indeterminate) How many indeterminate forms can you name? Did you know all 7? Math Wars!!! lim = ? Zero! Zip What are the three main types of discontinuities? 1. Hole at x=3 in the example 2. Vertical asymptote at x=1 in the example 3. Step usually the functions description is split up : for x0 f(x)= { 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 at x=a, then the derivative is undefined at x=a What is the definition of continuity at a point? What is a monotone function? A function that is either always increasing or always decreasing. (i.e. the derivative is always positive or always negative.) What is a normal line? The line perpendicular to the tangent line. Given (a,b) is on the graph of f(x) Did you remember that one? Its a bit esoteric, eh? 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, c, between a and b such that f(c) = p. 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. What does the Mean Value Theorem say? 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. What does f (a) = 0 tell you about the graph of f(x) ? Warning: irrelevant picture The graph has a horizontal tangent line at x=a. f(a) might be a minimum or maximumor perhaps just a horizontal inflection point. 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? The derivative must change signs at x=a What is the First Derivative Test? 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!! Whats the Second Derivative Test? 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 What do you know about the graph of f(x) if f (a) = 0 (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 f at x = a to confirm the inflection point actually occurs) How do you determine velocity? Velocity = the first derivative of the position function, or v(a) + (initial velocity + cumulative change in velocity) How do you determine speed? Speed = absolute value of velocity How do you determine acceleration? acceleration = first derivative of velocity = second derivative of position Using differentials to approximate f(a+h) with a point near (a,f(a)) on the tangent line what does f(a+h) ? This is driving me nuts!!!! f(a+h) f(a) + f (a) h Dont make an If f (x) is negative. Then f(x) is decreasing. If f (x) is positive. Then f(x) is increasing. If f (x) is negative then f(x) is concave down If f (x) is positive then f(x) is concave up How do you compute the average value of ? ______________________ b - a dx Note: This is also known as the Mean (average) Value Theorem for Integrals How do you locate and confirm vertical and horizontal asymptotes? 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 If = ky What does y = ? Calculus trivia: doubling time is = If you know life, how do you find k? Whats general formula for a Riemann Sum? ormore specifically Calculus trivia: as n (number of rectangles) goes to the summation sign becomes the integral sign and x becomes dx Whats the Trapezoidal Rule? The Trapezoidal Rule is the formula for estimating a definite integral with trapezoids. It is more accurate than a Riemann Sum which uses rectangles. Notice that all the y-values except the first and last are doubled. Use this only if the partitions are even, otherwise do them all by hand separately. Do we need to take a short break? Back already? What is LHopitals Rule? ^ Given that as x both f and g or both f and g then the limit of = the limit of as x LHopitals Rule: ^ What is the Fundamental Theorem of Calculus??? where F (x) = f(x) Do you know the other form? The one that is less commonly used? The FUN damental Theorem of Calculus: YAY!! Steves Theorem!! What is the general integral for computing volume by slicing? (Assume we are revolving f(x) about the x-axis) What if we revolve f(x) around y=a ? What if we revolve the area between 2 functions: f(x) and g(x) around the x-axis? Be sure to square the radii separately!!! (and put the larger function first) 1. How do you compute displacement? (distance between starting & ending points) 2. How do you compute total distance traveled? displacement: total distance: Yea!!! Thats all folks! (Be sure to check out the other calculus power point drill and practices)