AP Calculus BC Summer Assignment

Transcription

1 AP Calculus Chapter : Prerequisites for Calculus Test REVIEW AP Calculus BC Summer Assignment Name: Period: Note: Questions #-9 are non-calculator. Questions #0- are calculator. A B ) Consider ABC to the right: C C a. Write the equation of the line represented b segment AB in slope-intercept form. b. Write the equation of the line represented b segment BC in point-slope form. c. Write the equation of the line represented b segment CA in general form.

2 ) Let L represent the line - = 7. a. Write the point-slope form of the equation of the line through the point (, ) that is PARALLEL to line L. b. Write the slope-intercept form of the equation of the line through point (, 0) that is PERPENDICULAR to line L. ) Determine the value of "" for which the line through points A and B will have the given slope, m. A: (, 7) B: (-, ) m = - 7 ) Find the value of "k" that would make these two lines (a) PARALLEL and (b) PERPENDICULAR. - k = 6 + = - ) Geometric equations in function form. Write the area of an isosceles triangle with height 8 inches as a functions of one of its two equal sides.

3 6) Determine the DOMAIN of the following functions b algebraic intuition. Then based on that domain, determine the RANGE. A number line and/or graph ma be helpful. Write our answers in interval notation. a. f() = - - b. f() = - 9-7) Determine whether the following functions are EVEN, ODD, or NEITHER b means of an algebraic test. A graph ma also be helpful. a. = c. = - 7 b. = sin() d. = +

4 8) Graph the following piecewise function as accuratel as possible on the graph provided: +, - < - f() =, - -, < ) Given f() and g() below, find both the composite functions f g() and g f(). Then discuss an DOMAIN restrictions that each composite ma have. f() = + g() =

5 0) Graph the following eponential functions BY HAND. For full credit, ou must show a t-table and graph of the PARENT FUNCTION first. Then show an shifts to the final graph. Lastl, ou must list the DOMAIN and RANGE of the final graph. a. = b. = c. = e

6 ) Graph the curve represent b the given parameterized function on the aes provided below. For full credit, ou must show a t-table with the starting point, ending point and the direction of the curve with respect to time. = cos(t) = - sin(t) 0 t p ) Find a Cartesian equation for the parameterized curve shown below. (This happens to be the same parametric curve ou graphed b hand in the previous question.) = cos (t) = - sin(t) ) Find a parameterization for the line segment containing the endpoints ( -, ) and ( 8, ). Assume that the first point is where the curve begins and the second point occurs second later. Confirm ou answer b checking if our parameterization matches the same line segment in = m + b form. 6

7 ) Graph both f() and its inverse the "caveman wa" on the aes provided below. Based on our graphs, is f() a one-to-one function? How did ou determine this? Answer in a complete sentence. f() = ) For the function listed below, first find its inverse ALGEBRAICALLY. Then show that the inverse ou found is the correct inverse b wa of an algebraic PROOF. f() = ) Evaluate the following trigonometric values. Use trig triangles to help ou when necessar. Don't forget to pa attention to what quadrant ou're in! a. csc p 6 c. tan 7p b. cos - p d. cot p 7

8 7) Graph the following logarithmic functions BY HAND. For full credit, ou must show a t-table and grpah of the PARENT GRAPH first. Then show an shifts to the final graph. Lastl, ou must list the DOMAIN and RANGE of the final graph. a. = - log ( - ) b. = ln ( + ) ) PROVE that the following trigonometric identities are true. Show all algebra and identities step-b-step in the space provided below. a. sec () + - sec () - = - cot () b. tan () tan () + sec () = + csc () 8

9 9) Graph the following trigonometric functions BY HAND. For full credit, ou must CLEARLY show the period, amplitude, and phase shift fo each graph. Label everthing! a. = - csc - p 6 + b. = cos p + p 6 - c. = tan + p 9

10 0) You invest $,000 in an account that earns.6% interest compounded monthl. How long will it take for our investment to be worth $0,000? ) Radium has a half-life of 60 ears. If there were grams of Radium present initiall, how long will it take for it to deca down to grams? Use = Cekt. ) The table below shows the net profit (in thousands of dollars) of a furniture compan during various ears. Let = 0 represent the ear 900. Use our graphing calculator to find a logarithmic regression for the data, then use it to estimate in what ear the compan's profits reached 00 thousand dollars. Year Profit (in thousands of dollars)

11 ) A power plant is located on the south bank of a major river, at a point where the river is 600 feet wide. Power cables need to be installed for a town which is located. miles down stream, on the north bank. Cables laid under water cost $00 per foot, and cables that are laid on land cost $ per foot. a. The power compan is considering laing a cable straight across the river (to the north) and then straight down the river's edge on land to the town. How much would this cost? b. Reconsidering, the power compan now thinks it ma just la one large cable diagonall across the river (under water) from the power plant to the town. How much would this cost? c. The power compan now realizes that the least epensive option is probabl a combination of the two methods listed above. Write an algebraic epression for the total cost of laing the cables - part diagonall across the river, and part straight down the river's edge on land to the town. Then use our graphing calculator to graph the epression and find the MINIMUM value for "". Where EXACTLY should these cables be laid in order to minimize the cost?