PreCalculus Final Eam Review Revised Spring 0. f() is a function that generates the ordered pairs (0,0), (,) and (,-). a. If f () is an odd function, what are the coordinates of two other points found on the graph of f()? b. If f () is an even function, what are the coordinates of two other points found on the graph of f()?. Using Algebraic techniques, determine whether f ( ) = + is even, odd, or neither.. Given: f() = e and g() =. Find: g( + h ) g( ) a. g( f ( )) b. ( f g )( ) c. h d. f ( g( )) e. the average rate of change of g() between =0 and =. 0 0 < 0. Let f ( ) = + 0 0 0 a. State the domain of f (). Use an appropriate notation. b. Find f (). c. Graph this function. Clearl show our aes scales and label an important points.. The function D(p) = 00-00p gives the weekl demand for video rentals at Joe s Videolog when Joe charges p dollars to rent a video. a. What is the demand when Joe charges $.0 to rent a video? b. Find D - (0) c. Interpret D - (0) in contet of this problem.. The graph of function = is to be shifted horizontall units right and verticall units up. Write the function definition that represents this transformation.. Let C = f() be the circumference of a circle with radius cm. a. Eplain the meaning of C = f(+). b. Eplain the meaning of C = f() +.. The following table gives function values for f () and g() for =,,,,. Complete the table for g( ) m( ) = and n( ) = f ( g( )). Place an X in an boes where the value cannot be f ( ) determined. f() g() 0 m() n()
9. For the following functions: a. Find all the zeros and describe the long-run behavior of f ( ) = 0 +. b. Find all the zeros and describe the long-run behavior of f ( ) = + 0 9 + 0 0. Determine whether each table could represent a linear, eponential or periodic function and then find a possible formula for each. f() g() h() 0. 0. 0....... 9.. 0 9..0.. Using the figure shown below, match the formulas (i) - (vi) with a graph from (a) - (f). f ( ) (i) = f ( ) (ii) = f ( ) (iii) = f ( ) + (iv) = f ( ) (v) = f ( ) (vi) = f ( ) a. b. c. d. e. f.. Sketch a graph of a function = f( ) with all of the following features... o f (0) = o f ( ) = f () = 0 o f is decreasing for < 0 o f is increasing for > 0 o as, f ( ) o as, f ( )
g. Use the graphs of f and g below to find g (0), g( f ()), (fg)(-), ( ), (f + g)(-), (g - f )(-) f f () g(). Write a possible formula for the polnomials graphed below. Leave our answers in factored form. a. b. 9 0 9 9 0 9 0 9 0 9 9 0 9 0 c. 0 9 0 9 9 0 9 0 d. Describe the concavit of each function at =.
. State the domain and range of the following. a. = e b. = ln c. e. = = 9 d. = tan. Find the domain, -intercepts (if an), -intercepts (if an), asmptotes (if an), and long-run behavior. + + a. P() = b. Q() = c. R() = 9. Find the domain, range and an asmptotes for each of the following. a. = log ( ) b. = ln c. = e + d. =. Solve. Give our solutions in eact form. a. log + log(+ ) = b. ln( + ) = + c. e + = d. 0 + = 0, 000 9. The population of a cit has an annual growth rate of.% per ear. If the initial population of the cit is,000: a. Find a formula for P(t), the population in ear t. b. What will the population be in 0 ears? c. How man ears will it take the population to double? Round to the nearest ear. 0. A bank account earns a continuous interest rate of %. If $,000 is deposited into the account: a. Find a formula for B(t), the balance in the account in t ears. b. When will the balance reach $0,000? Give an eact answer and then round to the nearest tenth of a ear.. The graph at the right represents an eponential function of the form f ( t ) t = ab. Find a and b eactl. (0, ) (,.). Complete the problems which involve series. a. Evaluate the sum ( k ) 0 k= 0 b. Evaluate the sum () n= 0 n c. Epress the given series using compressed,, notation + + + ( ) + d. Epress the given series using compressed,, notation + + + 9
. Use the Binomial Theorem to epand ( +. Simplif completel.. a. Convert 0 to radians. b. Convert ) π to degrees.. Find the eact value of the si trigonometric functions of θ if cosθ = and θ is in Quadrant III.. Find the eact value of: a. tan sin b. sin cos c. sec tan. Solve each triangle. Round answers to one decimal place. a. A =, C = 90, b = b. A =, b =, c = c. A = 0, a =, b =. From a point on level ground feet from the base of a tower, the angle of elevation of the top of the tower is.. Approimate the height of the tower rounded to the nearest foot. 9. The angle at one corner of a triangular plot of ground is. and the sides that meet at this corner are feet and 0 feet long. Approimate the length of the third side rounded to the nearest foot. 0. For each function, identif the midline, amplitude, period, horizontal shift, and asmptotes, when appropriate. a. = cos b. = cos ( π ) c. = - sin d. = tan (). Find solutions for each equation. a. On the interval [0, π) give eact answers where possible. Otherwise, give answer correct to decimal places. i. sin θ sinθcos θ = 0 ii. sin + sin = 0 iii. sin () sin cos = - iv. tan sin = -sin ( eact answers, answers rounded to decimal places) b. Use our answers from part (a) to find all real solutions. π. A person s blood pressure, P, (in millimeters of mercur) is given b P = 00 0 cos t, where t is time in seconds. State the period, the midline and the amplitude and eplain the practical significance of these quantities.. Assume the graph below shows a portion of a sine curve. Find the amplitude, the period, the equation of the midline, and an horizontal shift. Net, write a function definition for the curve. Finall, state the domain and range for the function.
. Verif each identit: a. ( sin + cos ) = + sin( ) b. sin + cos = cos c. = cotθ cscθ cosθ + cosθ d. cos = sin + sin e. (cot + tan ) = sec + csc. Find the magnitude and direction of vector v= i+ j. Round answers to one decimal place.. A cclist rides mph due north as the wind blows him mph west. a. Draw a sketch of this situation using one horizontal vector and one vertical vector. b. On our sketch, draw the resultant vector which show the cclist s total displacement. Label this vector as v. c. Epress the resultant vector, v, from part (b) in component form. For vector v find the eact magnitude and direction correct to decimal place.
Answer Ke - PreCalculus Final Eam Review a. (-, -) and (-, ) b. (-, ) and (-, -). neither a. = b. ( f g )( ) = e ( ) g( f ( )) e e g( + h ) g( ) c. = + h d. h a. [0, 0] b. f () = c. a. D(.0) = 00 video rentals per week b. p = $. c. When the demand is 0 rentals weekl, the price is $... f ( ) = ( ) + f ( g( )) = e e. a. The circumference of a circle when the radius is increased b cm. b. The circumference of a circle is increased b cm. 0 0 0 0 00 0 0 0 0 0 (0, 90) (0, 0) (0,0) 90) 0 0 0 0 00 0 0 0. m() / / 0 / n() X 9a. {-, /, } as, and as -, - 9b. {-, -/, /, } as, and as -, 0. f ( ) =. ( ) π eponential; g( ) =. +. linear; h( ) =. sin +. periodic a. v b. vi c. ii d. iv e. i f. iii. Answers var 9 9 9 0
. a. c. g g ( 0 ) = ; g( f ( )) =. ; ( fg )( ) = ; ( ) = 0 f ( g f )( ) = ; ( f + g )( ) = ; = ( )( + )( ) b. = ( + )( ) = ( + )( )( ) d. a is concave up at =, b is concave down at =, c is concave down at = a. Domain (-, ) b. Domain (0, ) Range (0, ) Range (-, ) c. Domain (-, ) kπ d. Domain, k an odd integer Range [-9, ) Range (-, ) e. Domain (-, ) Range (-, ) a. Domain:, -int: (0., 0), -int: (0, -0.), VA: =, HA: =, as ±, b. Domain: { ± }, -int: (-0., 0), -int: (0, -), VA: =, = -, HA: = 0, as ±, 0 c. Domain: { ± }, -int: (0, 0), -int: (0, 0), VA: =, = -, as, and as -, - a. D = (, ) b. D = (0, ) c. D = (-, ) R = (-, ) R = (-, ) R = (, ) VA : = VA : = 0 HA : = d. D = (-, ) R = (-, 0) HA : = 0 = + d. = { ± } a. =, = -/ is etraneous b. = ( e ) c. ( ln ) 9a. P( t ) = 000(. 0 ) t 9b. P(0)=, people 9c. ears
0a. 0 0 = 000. t 0b. B( t ) e. a =, b =. ln0 t =. ears 0. 0 a. S 0 = 0 b. + S 0=, c. d. n= 0 n or n= 0 n= n + ( ) n or ( ) n There are man possibilities for c and d. n=. + 9 + + + π a. b. -0. sin θ =,cos θ =,tan θ =,csc θ =,sec θ =,cot θ = a. b. c. a. c =. a =. B = b. a =. B =.9 C = 0. c. B =. C =. c =.9 or B =. C =. c =.0. 0 ft 9. 9 ft + 0a. midline: = - Amp = per = π horizontal shift 0 0b. midline: = 0 Amp = per = π π horizontal shift right 0c. midline: = 0 Amp = per = π horizontal shift 0 π π kπ 0d. no midline per = VA at these -values + where k is an integer a. i. { 0,,.,. } π ii. iv. { 0, π,. 9,. 0} π π π,, iii. π π π,, b. i. { k,. k,. k } π + π + π ii. iii. π k, π k, π + π + π + kπ iv. π k, π k, π + π + π + kπ {kπ,.9 + kπ}. period = 0. sec midline: P = 00 amplitude = 0 mm of mercur Blood pressure fluctuates between a low of 0 and a high of 0 in 0. seconds.
. A = per = midline: = horiz shift: none f ( t ) = sin( π t ) + Domain: (-, ) Range: [-, ]. proofs - suggested first steps a. Square out left hand side then use a pthagorean identit. b. Use a pthagorean identit to replace sin with cos on the left-hand side. c. Find a common denominator on the left-hand side to combine the two fractions into one fraction. d. Use a pthagorean identit to replace cos with sin on the left-hand side. e. Square out the left-hand side.. v= v =. 9 θ = 0. a. b. v c. v = i + j, v= v =, θ =. west of north