You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need:

Size: px

Start display at page:

Download "You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need:"

Bernard Knight
5 years ago
Views:

1 You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need: Index cards Ring (so that you can put all of your flash cards together) Hole punch (to punch holes in your flash cards) Pen Color markers/pens/pencils Directions: First step: On the front of your index card you will write the left of the column. On the back you will write the right of the column. On the back side you will either draw an example or illustration with meaning to better understand the front of the flash card. Each flashcard should be numbered with the corresponding number. You will be graded on competition of all of the flash cards as well as how detailed your examples are. The last step is to create 5 flash cards of topics/concepts that aren t listed below that you feel would be useful information to know for your AP exam. Remember, you are being asked to rewrite the information below. DO NOT Print and cut the information onto a flash card.

2 1. What is a limit A limit is the y value of a function that x approaches from the left and right. ( the limit has to meet up) 2. What does it mean for a function to be continuous? f(a)exists lim 0 2 f(x) exists lim 0 2 f(x) exists = f(a) 3. What is Extreme Value theorem? 4. What are and how do you find points of Discountinuity? In a closed interval [a, b], f has a maximum and a minimum. (Create a table and check critical numbers and end point) Holes/Vertical asymptotes Factor out the equation and set the bottom equal to 0 5. How do you find Minimum/Maximum Values? 6. When and how do you apply L Hopitals Rule? First locate the critical numbers. Secondly create a number line to determine min/max. Lastly, plug in your x into the original to find y When you have an indeterminate form 9 9 or < <. Find the derivative of the numerator and denominator (separately), then plug in to find the limit. (repeat process if needed) 7. If lim 0 2 > lim 0 then Limit does not exist 8. Definition of a derivative f C (x) = lim D 9 f(x + h) f(x) h

3 9. Differentiability implies Continuity 10. Continuity does not imply Differentiability (ex. Sharp turn) 11. What can f be used for? 12. J J0 ( K L ) Slope Increasing / decreasing Instantaneous rate of change Slope of the line tangent Lo DHI hi DLO ( LO ) S g f fg ( g ) S 13. J J0 ( fg ) fg C + gf 14. J J0 f(g(x ) Chain rule f C Vg(x)W g C (x) 15. J J0 (sin(x)) Cos (x) (Starts with a c, so it s negative) 16. J (cos(x)) - Sin (x) J0 Write sec(x) sec(x) tan(x), then 17. J (Tan(x)) cover tan(x). You re left with J0 sec S x Write sec(x) sec(x) tan(x), then 18. J (sec(x)) cover sec(x). You re left with J0

4 sec x tan(x) (Starts with a c, so it s negative) 19. J J0 (csc(x)) Write csc(x) csc (x) cot(x), then cover one of the csc(x) and you re left with csc(x) cot(x) (Starts with a c, so it s negative) 20. J J0 (cot(x)) Write csc(x) csc (x) cot(x), then cover one of the cot(x) and you re left with csc S (x) One over the inside times the 21. J (ln(u)) derivative of the inside J0 22. ln (e) = ln (1) = ln (0) = 1 u u 1 0 Undefined 23. J J0 (ec ) e d u 24. J J0 (ac ) a d u ln a 25. J J0 (sinef u) u 1 u S 26. J J0 (cosef u) u 1 u S

5 27. J J0 (tanef u) u 1 + u S 28. If f ef (x) = g(x), then J J0 (g(x)) (derivative of the inverse) 1 f (g(x)) 29. Mean Value theorem If f is continuous on [a,b], and differentiable on (a,b), then for some c in (a,b), there is an f (c) = K(j)eK(2) je2 30. Average rate of change Average Velocity Average Acceleartion f(b) f(a) b a 31. How do you find critical numbers? Find the derivative and set it equal to zero. 32. How do you find vertical tangent lines Set the denominator of the derivative equal to zero 33. How do you find horizontal tangent lines(max/mins) Set the numerator of the derivative equal to zero. 34. How do you know if a function is at rest? f (x) = How do you know if a function is increasing? f (x) > 0

6 36. How do you know if a function is decreasing? f (x) < How do you know if a function is concave up? f (x) > How do you know if a function is concave down? f (x) < How do you know if a function has a maximum? The function changes from increasing to decreasing 40. How do you know if a function has a minimum? The function changes from decreasing to increasing 41. How do you know if a function has a point of inflection? The function changes from concave up to concave down or concave down to concave up. 42. Tangent lines at x = a y y f = f (a)(x x f )

7 43. Normal line at x = a Same as tangent line except opposite and reciprocal slope. y y f = 1 f (a) (x x f) 44. In calculus position is denoted by S(t), x(t) 45. In calculus velocity is denoted by S (t), x (t) 46. In calculus velocity is denoted by S (t), x (t) 47. How do you find speed? v(t) 48. How do you know if speed is increasing? V(t) and a(t) have the same sign 49. How do you know if speed is decreasing? V(t) and a(t) have the different signs 50. What is linear approximation Same as tangent line: y y f = f (a)(x x f )

8 51. If the graph of f is concave up, then the linear approximation is An underestimate 52. If the graph of f is concave down, then the linear approximation is An overestimate 53. When is a particle is going to the left? Velocity is negative 54. When is a particle is going to the right? When velocity is positive 55. What is the fundamental theorem of Calculus f(x)dx = j j o f(x)dx = F(b) F(a) 2 j - f(x)dx Second fundamental theorem of calculus J L(0) J0 2 f(t)dx = g (x) f(x) 58. Left Riemann Sum Create a chart 59. Right Riemann Sum Create a chart

9 60. Midpoint sum Create a chart 61. Trapezoidal Sum Create a chart 62. If you re asked to create a chart to find the area, how do you find how wide the points are? b a n n = the number of rectangles (subintervals) needed 63. If f is increasing on the interval [a,b], then a left Riemann sum is an Underestimate 64. If f is decreasing on the interval [a,b], then a left Riemann sum is an Overestimate 65. If f is increasing on the interval [a,b], then a right Riemann sum is an Overestimate

10 66. If f is decreasing on the interval [a,b], then a right Riemann sum is an Underestimate 67. how do you know if the tangent line is an overestimate or underestimate? (draw each example) Concave up = underestimate Concave down = overestimate 68. Total Distance (negative area turns to positive) j o f(x) dx = Area between two curves (Top curve bottom curve) j o f(x) g(x)dx = x s dx = Add one to the exponent and divide by the new exponent. 0 t@u svf +c 71. cos u du = Sin (u) + c 72. sin u du = - cos (u) + c

11 73. sec S u du = Tan(u) + c 74. sec(u) tan(u) du = Sec(u) + c 75. Jd = fed w sin ef u + c 76. Jd fvd w = tan ef u + c 77. e d du = (if there s one e then u is the exponent) e d + c 78. a d du = a d ln a + c 79. Jd d = ln u +c 80. Average amount (value) j 1 b a o f(x) dx If given the first derivative graph, where are my critical numbers? 82. If given the first derivative graph, what values are increasing? 83. If given the first derivative graph, what values are decreasing? They are on the x-axis Anything above the x-axis. Anything below the x-axis.

12 84. If given the first derivative graph, how do I find critical numbers of my second derivative? 85. If given the first derivative graph, where is it concave up? 86. If given the first derivative graph, where is it concave down? By locating max/mins on the first derivative graph. Wherever the first derivative graph is increasing. Wherever the first derivative graph is decreasing

Math 229 Mock Final Exam Solution

Name: Math 229 Mock Final Exam Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and that it