Quick Review for BC Calculus

Transcription

1 Quick Review for BC Calculus When you see the words This is what you should do 1) Find equation of the line tangent to at. 2) Find equation of the normal line to at (. 3) Given, where is increasing Set, make a chart and see where 4) Given a picture of, where is increasing? is increasing when the graph is above the x axis 5) Given, what is the local min? Find Set 6) Given, what is the local minimum? Set, make a chart and see where goes from neg to pos, make a chart and see where goes from neg to pos 7) Given a picture of, what is the local minimum 8) Given and is a minimum or maximum? 9) Given, and boundary [a, b], what is the absolute minimum? 10) Given, find the points of inflection. 11) Given find the points of inflection. 12) Given a picture of, find the points of inflection. 13) Given, find the points of inflection. Find the zeros of the graph and see where the graph goes from below the x axis to 0 to above the axis If, its concave up and a min If, its concave down and a max Find Set, make a chart and see where goes from neg to pos Check the endpoints Find. Find where. Make a chart. Find the values of x where changes signs. Find. Find where. Make a chart. Find the values of x where changes signs. Find where the slope of changes signs. Find the zeroes, make a chart and see where the sign changes. 14) Given, find the critical points Find where.

2 15) If area under from 0 to 3 16) Given a picture, does the limit exist? If the y coordinate from the right side of the x is the same as the y coordinate from the left side of the x, the limit exists. 17) Does the value have to exist for the No. It just needs to approach the same y value. limit to exist? 18) Given a picture, is the function continuous? 19) Given a piecewise function, is the function continuous? { 20) Given a piecewise function, is the function differentiable? { No holes, jumps or vertical asymptotes. Put the 2 into both equations. If you get the same y coordinate, it is continuous. Take the derivative of both equations. Put the 2 into both derivatives. If it is the same, it is differentiable at 2. 21) Find the horizontal asymptotes. Bill Gates (take the limit as x approaches 22) Find the horizontal tangent. Find where. 23) Find the vertical tangents. Find where is undefined. Find where the denominator of the derivative equals 0. 24) Average rate of change of on This is the slope between and 25) Average value of on 26) Instantaneous rate of change of at. 27) Given, find 28) Given, find. 29) Given, find. 30) Given, find.

3 31) Given, find where the particle goes left. Find where where, make a chart and see 32) Given, find speed. is speed 33) Given, is the particle speeding up? speeding up slowing down 34) 35) 36) 37) If is the inverse of, 38) Left Riemann sums Make rectangles using the left y values as the heights and the change in x values for the base 39) Find the area using midpoint rectangles Make rectangles using the midpoint y values as the heights and the change in x values for the base 40) Find area using trapezoids 41) is the antiderivative of 42)., is the antiderivative of, find 43) Approximate the value of by using the tangent line to at The h is the change in x values and the bases are the y coordinates Fun Theorem Find the tangent line at x = 0. Put in.1 for the x and solve for y. 44) ( ) 45) What do you need? 62% is a 5 50% is a 4 33% is a 3 17% is a 2 Have fun. Have fun. Have fun.

4 46) Given find Split it into two integrals. 47) Given a water tank with gallons initially being filled at the rate of gallons per minute and emptied at the rate of gallons/min, find the amount of water in the tank at minutes 48) Find the rate at which the water amount is changing at minutes for # 47 49) Find the time when the water is at a minimum for # 47 Let Solve for x. Make a chart and look for change (neg to pos). Check endpoints. 50) Find the area between curves & where and they intersect at ( ) 51) The area between and the x axis and Pancakes is called. Rotate about the x axis and find the volume. 52) Rotate the region about y axis or a vertical line You have to change You integrate with y values. 53) If Bill Gates 54) 55) Donut formula 56) Circumference of a circle Factor, cancel out the and then put the 3 in for the x. is the distance from the farthest curve to the axis, is the distance from the closest curve to the axis

5 57) 58) Graph the absolute value function and find the area under the curve. It makes two triangles. 59) 60) 61) u-substitution 62) u-substitution 63) Use logarithmic differentiation. Take the of both sides. 64) 65) 66) U-substitution Divide and conquer. Make two integrals. 67) Euler s method The other chart: 68) u-substitution 69) Is the approximation using a tangent line an Use the second derivative. over or under approximation to the real value? 70) 71.5) VooDoo ( )

6 71) Partial fractions If If 72) Tabular ) The position vector of a particle moving in the plane is ( ) Find the velocity. 74) Find the acceleration. 75) Find the speed. 76) Given the velocity vector, and position at find the position vector 77) F ind the area inside the polar curve 78) Write a series for centered at centered at ( )

7 centered at centered at Geometric 79) If, Find the sum. Geometric sum 80) Find the interval of convergence of a series. Use the ratio test. 81) converges if 82) Let be the sum of the first four terms of an alternating series for. Approximate. This is less than the fifth term of the series. 83) Given, find 84) Given, find 85) Given, find the arc length on. 86) Given, find the arc length. ( ) 87) Nth term test If the last term does not go to 0, the series diverges If the last term does go to 0, you know nothing. 88) Integral Test If the integral converges to a value, the series converges. If the integral diverges, the series diverges. 89) Ratio Test If It diverges 90) Direct Comparison If If It converges Test fails 91) Limit Comparison Compare the new function to a given converging or diverging series, a number are doing the same thing

8 92) Alternating Series The series converges if a) It alternates b) The terms are getting smaller absolutely c) The last term approaches 0 93) Geometric Series If the function can be written as if the series converges the series diverges If it converges, the series converges to. 94) P Series If the series looks like this if p > 1, the series converges If, the series diverges is the harmonic series which diverges 95) ( ) This is All of these are series. Are they converging or diverging? Give a reason. Harmonic seric Diverges Alternating series Converges P series Converges ( ) Geometric Converges Nth Term 5 Diverges Integral Test This diverges so the series diverges. Direct Comparison which is the harmonic series which diverges So diverges.