AP CALCULUS BC - FIRST SEMESTER EXAM REVIEW: Complete this review for five extra percentage points on the semester exam.
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1 AP CALCULUS BC - FIRST SEMESTER EXAM REVIEW: Complete this review for five etra percentage points on the semester eam. *Even though the eam will have a calculator active portion with 0 of the 8 questions, #7 on this review is the only problem that REQUIRES a calculator. The rest can theoretically be done without a calculator. While most can be done without a calculator, I still put an asterisk net to all the problems they may be on the calculator portion of the test. You need to be able to do or understand the following for the final eam:. Use acceleration, velocity, and position functions and charts to find average velocity, acceleration, displacement, and other characteristics of the object in motion.. Evaluate its by canceling, using conjugates, L Hospital s rule, and by looking at a graph.. Understand discontinuities and be able to identify them by looking at a function s graph and equation. 4. Find the equation of a tangent line to a curve at a given point. f ' at 5. Recognize the it definition of a derivative and be able to use it to find the value of a given point. 6. Generate the graph of f ' given the graph of 7. Generate the graph of f given the graph of ' f. f. 8. Use differentiation rules. 9. Find derivatives implicitly. 0. Use first and second derivatives to find characteristics of the function.. Solve related rates problems.. Solve optimization problems.. Find critical numbers of a function. 4. Mean Value Theorem 5. Etreme Value Theorem 6. Intermediate Value Theorem 7. Find antiderivatives (power rule, recognition, u-substitution, parts, and partial fractions). 8. Use Reimann Sums 9. Find linear approimations to a curve. 0. Find the derivative of an inverse function. Use the fundamental theorem of calculus. Problems: Evaluate the its in #-8: ln 4. tan
2 Evaluate the its in #-8: (continued) sin tan 7. / 8. e Use the graph of f below to answer #9-0. *9. f *0. f *. f *. f *. f *4. f *5. f *6. f *7. f *8. f *9. f *0. f f. *. Use its to find f '4 if *. Use its to find f ' if f. Find all points of discontinuity in #-5 and state which type they are:. f 4. f f if if 6. Find the equation of a line tangent to y 7 at the point, Find the equation of a line tangent to f at the point 7,. *8. Given the graph of f to the right, generate a graph for f '. 9. If f 7, find the intervals or coordinates where f a. is increasing e. has a local ma b. is decreasing f. has a local min c. is concave up g. has a point of inflection d. is concave down
3 Using the table, answer questions Time (seconds) Position (feet) *40. What is his average velocity over the interval [, 8]? *4. Over what -second interval is his average velocity the lowest? *4. Over what -second interval is his average velocity the highest? *4. Eplain why there must be a time on the interval 0t 6 such that the velocity of the object is 5 feet per second. 44 The displacement of an object is given by interval [, ]? s t t t. What s its average velocity over the Find the derivatives of the function in # f f f sin 48. f 4 8 f f ln cot 5. f log5 5. f 7 5. f Use the table to answer # f f ' g g' d d *55. *54. f g ; 4 d f d g ; d d *56. f g ; 4 *57. Let h g. Find ' h.
4 58. Find y ' if y 59. Find dy d if y y Find the equation of a line tangent to the curve in #58 at the point (, ). 6. Find the linear approimation to f at a =. *6. The length of a rectangle is increasing at a rate of 5 feet per second which the width is increasing at the rate of foot per second. When the length is 5 and the width is, how fast is the area increasing? *6. A seven foot ladder is leaning against a wall. You pull the base of the ladder away from the wall at a rate of ft/s. How fast is the top of the ladder sliding down the wall when it is 4 feet from the ground? 64. Find all critical numbers of the function f 4. *65. Find the absolute maimum and minimum values of the function f on the interval [0, ]. 66. Where is the graph f concave up? *67. A farmer has 0 feet of fence, and he wishes to make from it a rectangular pen for his pig, using a barn as one of the sides. In square feet, what is the maimum area possible for this pen? *68. A farmer wants to build a rectangular pen in his pasture for his cows. The fencing for the east and west sides costs $/yard and fencing for the north and south sides costs $5/yard. If he wants to enclose 600 square yards, what dimensions will minimize the cost of the pen? f on the closed interval, for the following. a. According to the mean value theorem, there must be a point c on, such that f c is equal to what number? *69. Consider the function ' b. What theorem states that there must be a point d on, c. What does the etreme value theorem say about f on this interval? such that f d 8?
5 Find the antiderivative of the functions in # f ' 7. f ' csc 7. f ' *7. Estimate the area under f sin from = to = 9 using 4 rectangles and a. right endpoints b. left endpoints c. midpoints 74. Find the eact area under f from = - to =. Find the following: 75. ln d 76. d 77. sin 4 d 78. arcsin d sin d 80. d cos cos 4 8. d 8. d 8. ln 4 5 d Find the f ' given the following integral defined functions f t dt 84. sec 4 f t t dt sin 85. f dt 86. t 4 f d 78
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