AP Calculus Summer Review Packet
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1 AP Calculus Summer Review Packet Name: Date began: Completed: **A Formula Sheet has been stapled to the back for your convenience!** anytime with questions: Complex Fractions When simplifying complex fractions, work towards one fraction in the numerator (you ll need to use common denominators for this) and one fraction in the denominator (using common denominators). Once you have fraction divided by fraction, you can multiply by the reciprocal. You ll multiply straight across, but should be able to cancel factors common to both the numerator and the denominator. 6 7( x 1) 6 7x 7 6 7x 1 7 x 1 x 1 x 1 x 1 1 7x 1 x 1 7x 1 = x = x x 1 x 1 x 1 x 1 x ( x ) x( x) x 8 x x x 8 x x x( x ) ( x )( x) x( x ) ( x )( x) x( x ) = 1 5( x ) 1 5x 0 1 5x 1 5 x( x ) 5x 1 x(5x 1) x ( x ) x x x x x x 8 x x x 8 Simplify each of the following a a 5 a. x 5 10 x. 1 x x x x Functions To evaluate a function for a given value, simply plug the value into the function for x. Recall: f og(x) f (g(x)) OR f [g(x)] read f of g of x Means to plug the inside function (in this case g(x) ) in for x in the outside function (in this case, f(x)). Given f (x) x 1 and g(x) x find f(g(x)). f (g(x)) f (x ) (x ) 1 (x 8x 16) 1 x 16x 1 f (g(x)) x 16x 1
2 Let f (x) x 1 and g(x) x 1. Find each.. f (t 1) 5. f g() Let f (x) sin x Find each exactly. 6. f 7. f Let f (x) x, g(x) x 5, and h(x) x 1. Find each. 8. h f () 9. f g(x 1) Find f (x h) f (x) h for the given function f. 10. f (x) 9x 11. f (x) 5 x Intercepts and Points of Intersection To find the x-intercepts, let y = 0 in your equation and solve. To find the y-intercepts, let x = 0 in your equation and solve. y x x x int. (Let y 0) 0 x x 0 (x )(x 1) x 1 or x x intercepts (1,0) and (,0) y int. (Let x 0) y 0 (0) y y intercept (0,) Find the x and y intercepts for each. 1. y x 5 1. y x x
3 Use substitution or elimination method to solve the system of equations. x y 16x 9 0 x y 9 0 Elimination Method Add the two equations together to eliminate the y-terms x 16x 0 0 x 8x15 0 ( x )( x 5) 0 x and x 5 Plug x = and x 5 into one of the original equations y y 9 0 y 0 16 y y 0 y Points of Intersection: (, 0), (5, ), and (5, ) Substitution Method Solve one equation for one variable. y x x 16 9 (1st equation solved for y) x ( x 16x 9) 9 0 Plug what y is equal to into second equation ( x x The rest is the same as x x previous example ) ( x )( x 5) 0 x or x 5 Find the point(s) of intersection of the graphs for the given equations. x y 8 x y x y 7 x y Interval Notation 16. Complete the table with the appropriate notation or graph. Solution Interval Notation Graph x 1,7 8 Solve the inequality. State your answer in BOTH interval notation and graphically. 17. x 1 0
4 Domain and Range Find the domain and range of each function. Write your answer in INTERVAL notation. 18. f (x) x f (x) x 0. f (x) sin x 1. f (x) x 1 Inverses To find the inverse of a function, simply switch the x and the y and solve for the new y value. f (x) x 1 Rewrite f(x) as y y = x 1 Switch x and y x = y 1 Solve for your new y Cube both sides x y 1 x y 1 y x 1 f 1 (x) x 1 Find the inverse for each function. Simplify Solve for y Rewrite in inverse notation. f (x) x 1. f (x) x Also, recall that to PROVE one function is an inverse of another function, you need to show that: f (g(x)) g( f (x)) x If: f (x) x 9 and g(x) x 9 show f(x) and g(x) are inverses of each other. f (g(x)) x 9 9 g( f (x)) x 9 9 x 9 9 x 9 9 x x x f (g(x)) g( f (x)) x therefore they are inverses of each other.
5 Prove f and g are inverses of each other.. f (x) x g(x) x 5. f (x) 9 x, x 0 g(x) 9 x Equation of a line Slope intercept form: y mx b Vertical line: x = c (slope is undefined) Point-slope form: y y 1 m(x x 1 ) Horizontal line: y = c (slope is 0) 6. Use slope-intercept form to find the equation of the line having a slope of and a y-intercept of Determine the equation of a line passing through the point (5, -) with an undefined slope. 8. Determine the equation of a line passing through the point (-, ) with a slope of Find the equation of a line passing through the point (, 8) and parallel to the line y 5 6 x Find the equation of a line perpendicular to the y- axis passing through the point (, 7). Radian and Degree Measure 180 o Use to get rid of radians and radians convert to degrees. radians Use to get rid of degrees and 180 o convert to radians. 1. Convert to degrees: a. 5 6 b. 5 c..6 radians 5
6 . Convert to radians: a. 5 o b. 17 o c. 7 Angles in Standard Position. Sketch the angle in standard position. a b. 0 o c. 5 d. 1.8 radians Reference Triangles. Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible. a. b. 5 c. d. 0 Unit Circle You can determine the sine or cosine of a quadrantal angle by using the unit circle. The x-coordinate of the circle is the cosine and the y-coordinate is the sine of the angle. sin90 o 1 cos 0 (0,1) (-1,0) (1,0) (0,-1) - 5. a.) sin180 o b.) cos70 o c.) sin(90 o ) d.) sin e.) cos60 o f.) cos() 6
7 Graphing Trig Functions fx = sinx fx = cos x y = sin x and y = cos x have a period of and an amplitude of 1. Use the parent graphs above to help you sketch a graph of the functions below. For f ( x) asin( bx c) d, a = amplitude, = period, b c = phase shift, and d = vertical shift. b Graph two complete periods of the function. 6. f (x) 5sin x 7. f (x) cos x Trigonometric Equations: Solve each of the equations for 0 x. Isolate the variable, sketch a reference triangle, find all the solutions within the given domain, 0 x. Remember to double the domain when solving for a double angle. Use trig identities, if needed, to rewrite the trig functions. (See formula sheet at the end of the packet.) 8. sin x 1 9. sinx 0. cos x 1 cos x 0 Inverse Trigonometric Functions: Recall: Inverse Trig Functions can be written in one of ways: arcsinx sin 1 x Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains. cos -1 x sin -1 x cos -1 x tan -1 x sin -1 x tan -1 x Express the value of y in radians. y arctan 1 Draw a reference triangle. NOTE: opposite & adjacent side lengths were labeled to create the tangent above, then missing hypotenuse was found using Pythagorean Theorem. 7-1
8 This means the reference angle is 0 or 6. BUT, y = 6 falls in the interval from y Answer: y = 6 For each of the following, express the value for y in radians. 1. y arcsin. y arctan(1) Find the value without a calculator. cos arctan 5 6 Draw the reference triangle in the correct quadrant first. 61 Find the missing side using Pythagorean Thm. 5 Find the ratio of the cosine of the reference triangle. cos For each of the following give the value without a calculator.. tan arccos. sin arctan sin 7 sin1 8 Vertical Asymptotes Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the x-value for which the function is undefined. That will be the vertical asymptote. 6. f (x) x x 7. f (x) x x (1 x) Horizontal Asymptotes Case I. Top Heavy There is no horizontal asymptote. (Slant asymptote, using long division, if numerator is exactly one degree larger than denominator) Case II. Same Degree The asymptote is the ratio of the lead coefficients. Case III. Bottom Heavy The asymptote is always y = 0. Determine all Horizontal Asymptotes. 8. f (x) x x 1 x x 7 9. f (x) 5x x 8 x x f (x) x5 x 7 8
9 Formulas to know and use TRIG: Reciprocal Identities: csc x 1 sin x sec x 1 cos x cot x 1 tan x Quotient Identities: tan x sin x cos x cot x cos x sin x Pythagorean Identities: sin x cos x 1 tan x 1 sec x 1 cot x csc x Double Angle Identities: LOGS: sinx sin xcosx tanx tan x 1 tan x cosx cos x sin x 1 sin x cos x 1 Logarithms: y log a x is equivalent to x a y Product property: Quotient property: Power property: Property of equality: Change of base formula: log b mn log b m log b n log b m n log b m log b n log b m p plog b m If log b m log b n, then m = n log a n log b n log b a LINES: Slope-intercept form: y mx b Point-slope form: y y 1 m(x x 1 ) Standard form: Ax + By + C = 0 9
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