AP Calculus Summer/Winter Assignment

Transcription

1 AP Calculus Summer/Winter Assignment Going into AP calculus, there are certain skills that have been taught to you over the previous years that we assume you have. If you do not have these skills, you will find that you will consistently get problems incorrect next year, even though you understand the calculus concepts. It is frustrating for students when they are tripped up by the algebra and not by the calculus. This packet is intended for you to brush up and possibly relearn these topics. We assume that you have basic skills in algebra. Being able to solve equations, work with algebraic expressions, and basic factoring, for example, should now be a part of you. If not, you would not be going on to AP calculus. The topics covered in the packet are skills that are used continually in AP calculus. The attached summer packet is for all students enrolled in AP Calculus in the fall or spring. This packet will be collected on the first day of school for each semester, late work will not be accepted, and will count as a grade. Rather than give you a textbook to remind you of the techniques necessary to solve the problem, we have given you several websites that have full instructions on the techniques. If and when you are unsure of how to attempt these problems, examine these websites. Don t fake you way through these problems. As stated, students are notoriously weak in them, even students who have achieved well prior to AP calculus. Use the websites. See other side of this paper Realize also that certain concepts are interrelated. Domain, for example, may require you to be expert at working with inequalities. Solving quadratic equations may involve techniques used in solving fractional equations. This packet is due the first day back in school in the fall or spring. It will be graded. You need to get off to a good start so spend some quality time on this packet this summer if you take a yearlong calculus or AP Cal AB fall semester one. If you take nd semester AP Cal AB, you have only.5 weeks of winter break. Do not rely on the calculator. Half of the AP exam next year is taken without a calculator. If you have any questions about any of these problems or techniques used in solving them, you may contact us amanda.bowels@cowetaschools.net; teresa.wilson@cowetaschools.net. Mathematically yours, ECHS AP Calculus Team

2 Below are listed topics in the review. You can certainly do Google searches for any of these topics Beginning algebra topics Exponents Negative and fractional exponents Intermediate algebra topics Domain Solving inequalities: absolute value Solving inequalities: quadratic Special Factoring formulas Function transformation Even and odd functions Rationalize numerator/denominator Advanced algebra topics Asymptotes Complex fractions Composition of functions Solving Rational (fractional) equations Properties of Logarithm Solving log & exponential equations Trigonometry Graphs Unit Circle Identities Solving Trig Equations Inverse Trig

3 AP Calculus Assignment Name *No Calculators *You must show ALL your steps to earn full credits on separate sheets of paper in order. A. For each function below listed below, find a) Domain and range in interval notation. b) Whether the function is even, odd, or neither c) Is a function One-to-one? If so, find its inverse d) Graph the function with labels. f (x) = x. f(x) = x.f(x) = x 4. f (x) = x 5. f (x) = x 6.f(x) = / x 7. f (x) = x 8. f(x) = s inx 9. f(x) = c osx 0. f (x) = a rc tanx. f (x) = t anx. f (x) = l nx. f (x) = e x 4. f (x) = x B. Multiplying Polynomials 5. (x ) 6. (x + ) 7. [(x ) + y] C. Simplifying Rational expression x 4x+49 x +4 x x+7 x+ + x x D. Rewrite the difference quotient by rationalizing the numerator. 0. ( x x )/. ( x + h x )/ h

4 . Solve for x (do not use a calculator): = a. 5 x+ = 5 b. c. log x = d. log x = log 4 4log 5. Simplify: a. log 5 + log (x ) log (x ) b. log 4 9 log c. log 5 d. log (0 0 ) e. ( x ) f. log 0 x + log 0 x x+ log Solve. Show all steps. Your answer should be exact value. a. e x+ = 7 b. e x 5e x + 6 = 0 c. e x e x = 0 50 d. = e. log (x x) 4+e x 4 = f. ln (5x ) = g. log (x + ) + log (x ) = log h. log 8 (x + 5) log 8 (x ) = i. log 6 (log 4 (log x)) = 0 j. log (log (log 5 5)) = x 5. The number of students in a school infected with the flu t days after exposure is modeled by the function, P (t) = 00. +e 4 t (a) How many students were infected after three days? (b) When will 00 students be infected? 6. Exponential growth is modeled by the function n = n ekt 0. A culture contains 500 bacteria when t = 0. After an hour, the number of bacteria is 00. (a) How many bacteria are there after four hours? (b) After how many hours will there be 8000 bacteria?

5 You should know the following identities: sin(-x)= - sin x cos(-x)= cos x sin x + cos x = sin(x)=sinxcosx cos(x)= cos x sin x = cos x = sin x cos(x ± y)=cosxcosy sinxsiny sin(x ± y)=sinxcosy ± cosxsiny Solve each equation in the interval [0, π ). Give exact values if possible. 7. s inx = 8. cos x = cosx 9. cosx + = sin x =. sin x + s inx =. s in (x) + s inx = 0. 4cos x 4 cosx = 4. sin x cos x = 0 5. c os (x) = 6. cos(x) = sinx Evaluate without a calculator 7. arcsin( ) 8. arccos(-) 9. arctan(-) 40. cos(arctan(5/6)) 4. tan(arcos(/)) 4. sec(arcsin(/)) 4. cot(arcos(x-))