Honors Calculus Summer Preparation 2018

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1 Honors Calculus Summer Preparation 08 Name: ARCHBISHOP CURLEY HIGH SCHOOL

2 Honors Calculus Summer Preparation 08 Honors Calculus Summer Work and List of Topical Understandings In order to be a successful student in the Honors Calculus course, it is imperative to practice and improve your current math skills. Calculus is a course in advanced algebra and geometry that is used to study how things change. To be successful in this course, students must be highly motivated, dedicated, capable learners who put forth great effort in working independently to fully understand the concepts in the honors calculus curriculum. The summer packet will act as a refresher of math concepts that should already be mastered when you enter the honors calculus course in September. This packet is a requirement for students entering honors calculus and is due the first week of class. Complete as much of this packet on your own as you can, then get together with a friend, me, or google the topic. As always, your best work is epected! Requirements for completing summer packet: You must complete each of the problems in the packet. You must show all of your work in the space provided on the packet or on separate paper, attached to the packet and labelled on each page. Be sure all problems are neatly organized and all writing is legible. In the event that you are unsure how to perform functions on your calculator, you may need to read through your calculator manual to understand the necessary synta or keystrokes. You must be familiar with certain built-in calculator functions such as finding values, intersection points, using tables, and zeros of a function. I epect you to come in with certain understandings that are prerequisite to Calculus. A list of these topical understandings is below. Summer Work Topics: Factoring Zeros/roots/-intercepts of rational and polynomial functions Polynomial Long Division Completing the square Write the equation of a line Quadratic formula Unit Circle Composite function and notation Solving trigonometric equations Domain/Range Interpreting and comprehending word problems Graphing, simplifying epressions, and solving equations of the following types: Trigonometric Rational Piecewise Logarithmic Eponential Polynomial Power Radical Honors Calculus Summer Prep 08 P a g e

3 Honors Calculus Summer Preparation 08 Topic : Properties of Eponents/Radicals PROPERTY Product of Powers a m a n = a m + n 4 = Power of a Power (a m ) n = a m n ( 4 ) = Power of a Product (ab) m = a m b m () = Negative Power a -n = n a (a 0) - = Zero Power a 0 = (a 0) 4 0 = Quotient of Powers = a m n (a 0) = Power of Quotient Fractional Eponents a a m n a b m = a b m m (b 0) = = y = y EXAMPLE 9 = 9 = ± 8 = 8 Radical Products a b = ab = 6 Radical Quotients Rationalizing radical denominators a b b = a b b b a b = a b a b = a a b + b b b Simplify: ) g 5 g ) (b 6 ) ) w -7 4) y y 8 5) ( 7 )(-5 - ) 6) (-4a -5 b 0 c) 7) -5y y 8) Epress the following in simplest radical form = = 5 9 = 5 9) 50 0) 4 ) 9 ) 69 ) Simplify Epress your answer using a single radical. Honors Calculus Summer Prep 08 P a g e

4 Honors Calculus Summer Preparation 08 Topic : Simplifying Comple Fractions Simplify: 4) 5) = 6) 4 = 7) n n n n = Honors Calculus Summer Prep 08 4 P a g e

5 Honors Calculus Summer Preparation 08 Topic : Complete Factorization of Polynomials Factor completely: 8) 5 9) ) ) 5 56 ) 8 7 ) 6 Topic A: Linear Equations and Inequalities 4) Write an equation of a line with slope and y-intercept 5. 5) Use point-slope form of a linear equation to find an equation of the line passing through the point (6, 5) with a slope of /. 6) Find an equation of a line passing through the points (-, 6) and (, ). 7 Find an equation of a line with an -intercept of (, 0) and a y-intercept of (0, ). Honors Calculus Summer Prep 08 5 P a g e

6 Honors Calculus Summer Preparation 08 Topic B: Quadratic Equations and Inequalities To find the verte from a quadratic equation in standard form: y a b c We use the formula: = b a Verte at: ( b, f ( b a a If a is > 0 then opens upward if a is < 0 then opens downward Quadratic equation in verte form: f() = a( h) + k Verte at (h, k) Quadratic Formula y = ( - root ) ( - root ) = b± b 4ac a 8. Find an equation for the parabola whose verte is (, 5) Epress your answer in the verte form for a quadratic. and passes through (4,7). 9. Solve the inequality: 0. A. (, 4) (, ) B. 4, C. (,4) D. (, ) (4, ) 0) Transform y = to verte form by completing the square. Topic 4: Polynomial Functions. Which of the following could represent a complete graph of number? f ( ) a, where a is a real A. B. C. D. ) Find a degree polynomial with zeros -,, and 5 and going through the point (0, ). ) Use polynomial long division to rewrite the epression ) Use a graphing calculator to approimate all of the function s real zeros. Round your results to four decimal places. f 6 5 ( ) 5 4 Topic 5: Graphs of Parent Functions Honors Calculus Summer Prep 08 6 P a g e

7 Honors Calculus Summer Preparation 08 5) Graph each function: f() = f() = Domain Range Domain Range f() = f() = Domain Range Domain Range Honors Calculus Summer Prep 08 7 P a g e

8 Honors Calculus Summer Preparation 08 f() = f() = Domain Range Domain Range f() = f() = log Domain Range Domain Range Honors Calculus Summer Prep 08 8 P a g e

9 Honors Calculus Summer Preparation 08 Topic 6: Rational Functions and Asymptotes A rational function is a function of the form: p R q Where p and q are functions The Domain of a Rational Function We need to make sure that the denominator is not equal to zero. p R q q() 0 Eamples: 5 R H F , -4, - The graph looks like: f Since the graph is never equal to zero, the graph gets closer and closer to = 0 but never touches the y-ais. This line is called a vertical asymptote We compare the degree in the numerator and in the denominator to tell us about horizontal asymptotes. If the degree of the numerator is less than the degree of the denominator, 5 R o y = 0 is a horizontal asymptote. 4 If the degree of the numerator is to the degree of the denominator, then there is a horizontal coefficient of lead term of numerator asymptote at the line y = Eample: R coefficient of lead term of denominator 45 = 4 = So there is a horizontal asymptote at y = Oblique Asymptote: If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote, but an oblique (slant) one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder. R Eample: 5 4 = ( + 5)remainder (8+5) ( +4) Slant asymptote is at y = +5 Honors Calculus Summer Prep 08 9 P a g e

10 Honors Calculus Summer Preparation 08 6) Graph the following rational functions. Label all asymptotes and/or holes in the graph. Identify the key points of the graph. f() = Domain Intercepts Vertical Asymptotes Horizontal Asymptotes f() = 4 Domain Intercepts Vertical Asymptotes Horizontal Asymptotes 7) Give that f( ) 5. Find the asymptotes and the domain of the function. Domain: Vertical Asymptote(s): Horizontal Asymptote(s): 8) Factor to solve the inequality. Write your answer in interval notation Honors Calculus Summer Prep 08 0 P a g e

11 Honors Calculus Summer Preparation 08 Topic 7: Function Composition and Inverses 9) Let f ( ) and g. Compute ( )( ), ( ) g f 40) Find f(g()) if f() = and g() = + 4 4) Let 7 f( ). Find f ( ), the inverse of f( ). 4) Find f () if f() = + 6 Honors Calculus Summer Prep 08 P a g e

12 Honors Calculus Summer Preparation 08 Topic 8: Logarithmic and Eponential Functions Eponential Functions: f() = a b where b is the base and is the eponent. If the bases are equal, then b b Eponential Growth and Decay: When your growth factor is constant f() = a( + r) and f() = a( r). Compound interest: A() = P( + r n )nt Continuous compounding interest : A() = Pe rt In general terms: A(t) = A 0 (e kt ) where k is a constant of growth or decay 4) The graph of y for a a is best represented by which graph? A. B. C. D. 44) Rewrite the epression 4 log 5( ) y into an equivalent epression by epanding. 45) Solve: log 6 ( + ) + log 6 ( + 4) = Honors Calculus Summer Prep 08 P a g e

13 Honors Calculus Summer Preparation 08 46) Solve: log - log 00 = log 47) The number of elk after years in a state park is modeled by the function 6 Pt (). 0.0t 75e a. What was the initial population of elk? t b. When will the number of elk be 750? 48) Arturo invests $700 in a savings account that pay 9% interest, compounded quarterly. If there are no other transactions, when will his balance reach $4550? Topic 9: Trigonometry Honors Calculus Summer Prep 08 P a g e

14 Honors Calculus Summer Preparation 08 Sketch the graphs using the intercepts, amplitude, period, frequency and midline. 49). f( ) sin Intercepts: Amplitude: Period: Frequency: Midline: 50). f( ) cos( Intercepts: Amplitude: Period: Frequency: Midline: Honors Calculus Summer Prep 08 4 P a g e

15 Honors Calculus Summer Preparation 08 5) Find the eact value of each without the use of a calculator. sin a) b) cos c) 5 tan 6 d) csc e) cot 5) Find the perimeter of a 0 f) sin = slice of cheesecake if the radius of the cheesecake is 8 inches. 5) Two students are 80 feet apart on opposite sides of a telephone pole. The angles of elevation from the students to the top of the pole are 5 and. Find the height of the telephone pole. 54) Find the complement of. 55) Determine the quadrant in which a 0 angle lies. 56) Rewrite in radian measure. Round to three decimal places. 57) Rewrite in degree measure. Round to three decimal places. 58) After leaving the runway, a plane's angle of ascent is and its speed is 67 feet per second. How many minutes will it take for the airplane to climb to a height of,500 feet? Round answer to two decimal places. Honors Calculus Summer Prep 08 5 P a g e

16 Honors Calculus Summer Preparation 08 Topic 9B: Trigonometric Identities and Equations Law of Sines Law of Cosines csc( ) tan( ) sin( )cos( ) 59) Simplify A. C. sin( ) cos ( ) B. sin ( ) cos( ) D. cos( ) sin ( ) cos ( ) sin( ) 60) Solve the equation sin ( )cos( ) cos( ) algebraically. 6) Find all the eact solutions to sin ( ) sin( ) 0 0,. on the interval Honors Calculus Summer Prep 08 6 P a g e

17 Honors Calculus Summer Preparation 08 6) Juan and Romella are standing at the seashore 0 miles apart. The coastline is a straight line between them. Both can see the same ship in the water. The angle between the coastline and the line between the ship and Juan is 5 degrees. The angle between the coastline and the line between the ship and Romella is 45 degrees. How far is the ship from Juan? Miscellaneous Problems: 4 6) Determine the domain, range and the zeros of: f ( ) 0. 64) Solve the equation both algebraically and graphically Algebraic Solution: Graph Solution: 65) Three sides of a fence and an eisting wall form a rectangular enclosure. The total length of a fence used for the three sides is 40 ft. Let be the length of two sides perpendicular to the wall as shown. Write an equation of area A of the enclosure as a function of the length of the rectangular area as shown in the above figure. Then find value(s) of for which the area is 5500 ft? Eisting wall Honors Calculus Summer Prep 08 7 P a g e

18 Honors Calculus Summer Preparation 08 66) Describe the transformations that can be used to transform the graph of f ( ) graph of f ( ) ( ). Graph each function in the designated area below. to a f ( ) f ( ) ( ) 67) Graph the piecewise function. f ( ) 5 68) For the function f( ) graphed answer the following A. C. f () f (0) B. f ( ) 0 D. f ( ) ) Simplify the epression as much as possible. 0 ( 4) 6 ( 4) ) Use algebra to find the eact solution to = 0. Show all work. Honors Calculus Summer Prep 08 8 P a g e