AP Calculus Summer Prep

Transcription

1 AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have this packet completed for the first day of school and to not use as graphing calculator unless noted. Questions will be answered on the first day of class. Please note those problems that were challenging in the space below. NOTES:

2 ALGEBRA TOPICS FACTORING POLYNOMIALS Perfect Square Trinomials: a ab b ( a b) a ab b ( a b) Difference of Squares: a b ( a b)( a b) Sum of Cubes: a b ( a b)( a ab b ) Difference of Cubes: a b ( a b)( a ab b ) PRACTICE PROBLEMS Factor completely: ) x x 0 ) 5x 6y ) x(y ) + ( y) 4) x ( ) (x + ) + x ( ) (x + ) SOLVING POLYNOMIAL INEQUALITIES Example : Solution: x 4x 5 0 x x 5 0 factor and determine critical values: x = -, Answer: (-, 5) mark the zeros; pick a test point to determine the sign of the polynomial in each interval- this is called a SIGN CHART! PRACTICE PROBLEMS Include a Sign Chart 5) x + 7x + 0x > 0 6) x 6x

3 PRE-CALCULUS TOPICS Analyzing Graphs: Domain and Range, x-intercepts (zeros) and y-intercepts, extrema (local and absolute) End Behavior: lim x ± f(x) Continuity: removable, non-removable, All Polynomials are continuous for all x. Even/Odd Functions: Even: Symmetric to the y-axis. Algebraically: f(-x) = f(x) Odd: Symmetric to the origin. Algebraically: f(-x) = f(x) PRACTICE PROBLEMS (Noncalculator) Analyze the following functions without graphing them on a graphing calculator (except where noted). 7) f(x) = x 0x ) g(x) = x 6x Domain: Range: x-intercept(s): Domain: Range: x-intercept(s): y-intercept(s): y-intercept(s): Extrema (by hand): Extrema (on calc): End Behavior: End Behavior: lim x f(x) = lim x g(x) = lim x f(x) = lim x g(x) = Interval of Continuity: Interval of Continuity: Tests for Symmetry: Tests for Symmetry:

4 Rational Expressions and Functions: Domain and Range End Behavior: lim x ± f(x) Asymptotes: vertical and horizontal asymptotes Simplify 8 9) Consider f( x) x. Domain: x-intercept(s): Range: y-intercept(s): End Behavior: lim x f(x) = lim x f(x) = Interval of Continuity: Tests for Symmetry: 0 Simplify. x x x 0) hx ( ) x x x ) x 5x 6 x 4 ) x y x y 4 4 x y ) 5x(x +) (x+) +0(x+)(x +) 0(x+)(x +) 4

5 FUNCTION OPERATIONS AND COMPOSITION; INVERSE FUNCTIONS Composition of a function g with a function f is defined as: h( x ) g( f ( x )) The domain of h is the set of all x-values such that x is in the domain of f and f(x) is in the domain of g. Inverses: Functions f and g are inverses of each other provided: f ( g( x )) x and g( f ( x )) x f The function g is denoted as, read as f inverse. Horizontal Line Test: If any horizontal line which is drawn through the graph of a function f intersects the graph no more than once, then f is said to be a one-to-one function and has an inverse. Let f and g be functions whose values are given by the table below. Assume g is one-to-one with inverse g. X f(x) g(x) ) f( g ()) 5) g (4) 6) f( g (6)) 7) f ( f ( g())) 8) g( g ()) 9) If f(x) = x x + and g(x) is the inverse of f(x). Determine g(). 5

6 TRIGONOMETRY Trig values for selected angles: Radians Degrees Sin x Cos x Tan x Und. Key Trig Identities formulas you should know: sin x + cos x = sin(x) = sin x cos x cos(x) = sin x + tan x = sec x sin( x) = sin x ODD cos( x) = cos x EVEN Without using a calculator or table, find each value: 0) cos ( π ) ) sin (π ) ) cos 6 (5π) 4 Solve the trigonometric equations algebraically by using identities and without the use of a calculator. You must find all solutions in the interval0 : ) sin 0 4) cos x tan x cos x 5) 6) sinx cos x cos x cosx 0 6

7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Logarithm: For any positive numbers b and y with b, we define the logarithm of y with base b as x follows: log b y = x if and only if b y LAWS OF LOGARITHMS (for M, N, b 0, b ) ( i ) log MN log M log N M ( ii ) logb logb M logb N N ( iii ) log M log N if and only if M N k ( iv ) log M k log M logb M ( ) b b b b b v b M b b The logarithmic and exponential functions are inverse functions. Example: Consider f( x) x and g( x ) log x. Verify that (,8) is on the graph of f and (8,) on the log graph of g. In addition, ( ( )) x x f g x x and g( f ( x )) log ( ) x which verifies that f and g are indeed inverse functions. Simplify. log 8) 5 7) 5 log 4 9) 4 4 log 5 5 0) t ln et ) e ln(x+5) ) e lnx ) 4 ln e x 4) e x+ln5 (ln(x)+ln 5) 5) e Write the given expression as a rational number or as a single logarithm. 6) 5 ln (x) ln (y) + ln (4x) 6 ln (y ) 7

8 Limits: lim x c f(x) = N Limit as x approaches c of f(x) equals N LIMIT CONTINUITY Understanding the difference between limits and continuity: Limit is the value (y) that the function APPROACHES as you get close to c (either from one side or from both sides). Continuity implies that the limit exists (from both sides) and that the limit = the value at c! Limit Exists: lim x c + f(x) = lim x c f(x) Continuous: lim x c + f(x) = lim x c f(x) AND = f(c) = A (where A is a constant not ± ) One-Sided limit: exists if graph approaches a value from one side: + (right) or (left) Evaluating Limits:. Tables: either from table or by graphing and viewing table looking for values of y approaching the same number. Check one sided limits first and if they are equal the overall limit exists.. Graphically: Check one sided limits (again value of limit is the y-value or the height of the graph) and if they are equal overall limit exists.. Algebraically: plug in c. If you get a constant then limit exists. Rational functions may need to be simplified first! Properties:. lim x c A = A : the limit of a constant A is the constant A- think of it as the graph of a horizontal line. Y = A independent of what the x value is.. lim x c [f(x) ± g(x)] = lim x c f(x) lim x c g(x) : you can separate sum and difference of terms.. lim x c A(f(x)) = A lim x c f(x): you can pull out numeric factors. 4. lim x c (f(x)) n = [lim x c f(x)] n you can evaluate limit first, then raise your solution to the power. 5. lim x c [ f(x) ] = lim x c f(x) you can evaluate the limit of the numerator and denominator separately then g(x) lim x c g(x) evaluate their quotient. Examples: Evaluate the limits: 7) lim x (x ) 8) lim x + x 4 9) lim π x (x sin x) x, if x < 40) If f(x) = {, if x =, determine the following: 5x, if x > a. lim x f(x) b. lim x + f(x) c. lim x f(x) d. Is f continuous? 4) If lim x c f(x) = and lim x c g(x) = 5, determine the following: a. lim x c 5f(x) b. lim x c [f(x) g(x)] c. lim x c f(x) g(x) 8

9 4) Let f(x) be the graph below. Determine the following: a. lim x f(x) b. lim x + f(x) c. lim x f(x) d. f(-) e. Is f continuous at -? f. lim x f(x) g. lim x + f(x) h. lim x f(x) i. f(-) j. Is f continuous -? k. lim x f(x) l. lim x + f(x) m. lim x f(x) n. f() o. Is f continuous at? p. lim x f(x) q. lim x + f(x) r. lim x f(x) s. f() t. Is f continuous at? u. lim x f(x) v. lim x f(x) 9

10 Secant and Tangent Lines: Average Rate of Change: slope through points. Secant Line: line connecting points on f(x). m sec = f(x ) f(x ) x x Instantaneous Rate of Change: slope at a single point on f(x). Tangent line: line through a single point with slope approximated at that point. f(x + h) f(x) m tangent = lim h 0 h 4) Find the instantaneous rate of change of y = 4x at x = 44) Find the average rate of change of y = 4x between x = 0 and x = 45) Determine the slope of the secant line on the curve f(x) = x + at x = and x = and the slope of the tangent line at x =. Compare your slopes by sketching f(x), the secant line through x = and x = and the tangent line at x =. 46) Determine the slope of the secant line on the curve f(x) = x at x = and x = and the slope of the tangent line at x =.5. Compare your slopes by sketching f(x), the secant line through x = and x = and the tangent line at x =.5. 0

11 ANSWER KEY. (x + 5)(x ). (5x + 4y)(5x 4y). (x )(y ) 4. x( ) (x+4)(x+) (x+) 5. Sign Chart: -5-0 Solution: ( 5, ) (0, ) 6. Sign Chart: Solution: (, 4] [0, 4] 7. f(x) = (x 5) + 7 Domain: (, ) Range: [7, ) x-intercept(s): none y-intercept(s): (0, 57) Extrema (by hand): absolute minimum: (5, 7) End Behavior: lim x f(x) = lim x f(x) = Interval of Continuity: (, ) Tests for Symmetry: f( x) = x + 0x + 57 neither 8. g(x) = x(x ) Domain: (, ) Range: (, ) x-intercept(s): (0, 0), (, 0), (, 0) y-intercept(s): (0, 0) Extrema (on calc): local max:.66 local min: -.66 End Behavior: lim x g(x) = lim x g(x) = Interval of Continuity: (, ) Tests for Symmetry: g( x) = x + 6x = g(x)odd 9. Domain: (, ) Range: (0, 4] x-intercept(s): none y-intercept(s): (0,4) End Behavior: lim x 8 +x = 0 lim x 8 +x = 0 Interval of Continuity: (, ) Tests for Symmetry: f( x) = 8 +x = f(x)even h(x) = x+ x.. 5x (x )(x )(x+) (x+y)(x +y ). x +x +9x+8 4(x+) (x +) ½. - ½.. θ = π 4, π 4, 5π 4, 7π 4 4. x = π 4, 5π 4 5. x = π, π, 5π 6. x = π, π, π 6, 5π ¼ (x + 5). x. 4x 4. 5e x 5. 5x

12 6. ln 6x7 y m sec = f() f() = 4 (x+h) m tangent = lim + (x +) h 0 h m tangent at x = is Secant Line = x 9. π 40. a. -5 b. -5 c. -5 d. NO 4. a. -0 b. -7 Tangent Line c a. - b. - c. - d. DNE e. NO f. 4 g. 4 h. 4 i. j. NO k. - l. m. DNE n. DNE o. NO p. q. - r. DNE s. t. NO u. 4 v. - 4(x+h) 4. lim 4x h 0 = 4x lim +8xh+4h 4x h 0 h At x = : 8() = 6 h 8xh+4h = lim h 0 = 8x h 46. m sec = f() f() = 9 (x+h) m tangent = lim x h 0 = 6x m tangent at x =.5 is 9 Secant Line h Tangent Line 44. f() f(0) 0 = 4( ) 4(0 ) = 8