Math 143 Final Review - Version B page 1

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1 Math Final Review - Version B age. Simlif each of the following. cos a) + sin cos (log = log ) c) log (log ) log d) log log log e) cos sin cos f) sin cos + cos sin g) log sin h) sin tan i) + tan log j) log log log k) cos sin l) sin cos m) log log + log () n) log log + log o) log log + log. For each of the following functions given, sketch its grah and state its domain and range. a) f () = sin g () = cos c) h () = tan. Comute each of the following. a) sin (: ) (Hint: : is half of what angle?) c) cos tan if sin = and is in the rst quadrant.. Simlif each of the following. a) tan sin ( )!! c) sin cos d) cos sin e) cos cos f) cos cos g) sin cos h) cos tan i) sin tan j) tan sin k) sin cos l) cos tan ( ) m) tan tan ( ) n) sin sin + cos o) sin cos ) cos tan q) sin tan. Prove each of the following. a) sin + sin = cos cos cos = sin c) + tan tan =. Comute the inverse for each of the following functions. cos ( ) cos cos a) f () = f () = + c) f () = ( ) + d) f () = e) f () = + f) f () = e g) f () = log ( ). Grah the function given and grah the inverse relation in the same coordinate sstem. You do not have to nd the equation for the inverse. a) f () = log f () = jj c) =

2 Math Final Review - Version B age. Grah each of the following. State the stes in grahing the functions. a) f () = g () = + + c) h () = log + ( ). Grah each of the following functions b rst aling a division and then transformations on the grah of =. List the stes in grahing each function. a) f () = f () = +. Grah each of the air of functions in the same coordinate sstem. c) f () = a) f () = sin and g () = csc f () = sin and g () = sin c) f () = sin and g () = sin () d) f () = cos and g () = cos + e) f () = cos and g () = cos. If we know that sin = sin + and that cos <, then nd the eact value of sin.. Find the sum cos + cos + ::: + cos. Perform each of the following divisions. a) + ( + ) c) + ( ) d) + e) ( + ) ( ) f) ( ) ( + ) g) + h) ( ). a) Solve the equation + = if we know that is a solution of the equation. Solve the equation = :. Suose that log = m. Write log in terms of m.. Solve each of the following ineqalities. a) +. Solve each of the following equations. a) log ( + ) + log ( ) = log ( ) = c) + = d) log + + log ( + ) = e) sin cos = f) log ( + ) + log ( + ) = g) log + log = c) d) > h) log log = i) + = + + j) = + k) sin cos = l) sin = m) sin cos = n) sin cos =. Find the smallest and greatest value of the function f () = sin + cos. Find an equation for the tangent line drawn to the circle ( ) + ( + ) = at the oint (; ).

3 Math Final Review - Version B age. Comute the eact value of each of the following. a) sin cos c) cos tan ( ) sin cos d) tan tan + tan e) sin sin + cos f) cos sin. Grah each of the following functions. a) f () = g () = + c) h () = ( + ) ( ) d) f () = ( + ) ( ) ( ) e) f () = ( + ) ( ) f) f () = ( + ) ( ). Grah = f () given the grah of = f (). a) c) d) Given the grah of a function = f (), grah = f () in the same coordinate sstem Given the same grahs as in the revious roblem, grah the inverse relation in the same coordinate sstem.. The number of cells in a samle at time t (measured in hours) is N (t) = : :t : a) How man cells are in the samle at t =? How long will it take for the samle to double from the amount that it had at t =? c) How man cells are in the samle at t =? d) How long will it take for the samle to double from the amount that it had at t =? e) What do ou observe? Can we make (and erhas rove) a general statement?. Is there a right triangle whose sides are consecutive integers?. Write log log as a single logrithm.

4 Math Final Review - Version B age. Classif the discontinuities of each of the following functions as a hole or a vertical asmtote. a) f () = g () = ( + ) ( + ) ( ) ( ) ( ) ( + ) ( + ) ( ) ( ) tan. Prove the identit = sin tan + + sin. Find the domain for each of the following functions. ln ( + ) a) f () = ln ( ) + g () = ln c) h () = sin cos d) h () = cos sin sin e) f () = ln sin f) f () = sin cos g) f () = log (sin + cos ). Comute the eact value of each of the following. a) sin if cos = cos if cos = c) cos if sin = d) sin if tan = e) Comute the eact value of tan if we know that and cos =.. Solve each of the following triangles. a) a = m, c = m, = a = m, c = m, = d) a = ; b = ; and = f) a = ; b = ; and = c) a = m, c = m, = e) a = ; b = ; and =. a) Find the eact value of the cosine of the smallest angle in a triangle with sides ; ; and. Find the eact value of the cosine of the largest angle in a triangle with sides cm, cm, and cm long.. Find the eact value of the area of a triangle with sides ; ; and.. Two sides of a triangle are ft and ft long. Find the eact value of the third side if we know that the area of the triangle is ft.. Triangle SML has sides of length ; ; and : Find the eact value of cos S + cos M + cos L.. a) Find the eact value of cos if we know that cos =. Find the eact value of sin if we know that sin =.. Find tan if we know that tan = and tan ( + ) =.. A triangle has sides of length a, b, and c; which are consecutive integers in increasing order, and cos = : Find cos.

5 Math Final Review - Version B age. Consider an equilateral triangle with sides unit long, inscribed in a circle. Let O be the center of the circle. a) Find the radius of the circle. Find the distance between O and the side of the triangle.. The oulation of a town is growing eonentiall. From ; it took A ears for the oulation to double. From ; it took B ears for the oulation to trile. Eress B in terms of A: Answers. a) C) d) e) f) g) h) i) j) k) l) m) n) o). see handout r +. a). a) j) k). a) sin + sin = cos c) c) d) l) m) e) n) f) g) h) o) ) + i) q) + sin + sin = sin ( + ) + sin ( ) = sin cos + cos sin + sin cos cos sin = sin cos = cos = cos cos cos = sin c) + tan tan = cos cos = cos ( ) cos ( + ) cos ( ) cos cos LHS = + tan tan = + = cos ( ) cos cos = RHS = cos cos + sin sin (cos cos sin sin ) = cos cos + sin sin cos cos + sin sin = sin sin = sin = sin sin sin cos cos cos cos sin sin = + cos cos cos cos = cos cos + sin sin cos cos

6 Math Final Review - Version B age. a) f () = + f () = c) f () = + d) f () = e) f + () = f) f () = (ln ( + ) + ) g) f () = ( + ). a) f () = f () = jj c) = a) f () = + g () = + + c) h () = log ( ) Start with = Start with = Start with = log shift to the left b units re ect to the ais shift to the right b unit stretch along b the ais b shift to the right b units stretch along b the ais b re ect to the ais re ect to the ais re ect to the ais shift u b units shift u b unit shift down b units a) f () = = f () = + c) f () = + = + + = + Start with = Start with = Start with = shift to the right b unit shift to the left b units shift to the right b units stretch along b the ais b stretch along b the ais b stretch along b the ais b re ect to the ais re ect to the ais shift u b units shift u b units shift u b units

7 Math Final Review - Version B age. a) f () = sin and g () = csc f () = sin and g () = sin c) f () = sin and g () = sin () d) f () = cos and g () = cos + e) f () = cos and g () = cos a) + R + R c) + + R d) + R e) R f) R g) + R + h) R. a) ; ; ; ; ; ;. m +. a) ( ; ] log = log log = log + log ; = + log = + m = m + [ (; ) c) (; ] [ ( ; ] d) ( ; ) [ (; ). a) no solution c) d) ; e) + k, + k where k Z f) g) ; h) i) log j) k) + k + k + k where k Z l) = + k or = + k where k Z m) + k or + k where k Z n) + k. smallest value: greatest value:. a) = +. a) or + k where k Z = and = c) = and = c) d) e) f)

8 Math Final Review - Version B age. a) f () = g () = + c) h () = ( + ) ( ) d) f () = ( + ) ( ) ( ) e) f () = ( + ) ( ) f) f () = ( + ) ( ). a) c) d)

9 Math Final Review - Version B age ln :. a) : hours c) d) : hours : ln : e) For an time t, we will need to wait until t + ln for the samle to double. So, the doubling time is ln : indeendent of t and is constant. Let t be an time and t is the time when the amount is doubled from A (t ). In other words, A (t ) = A (t ). A (t ) = A (t ) : :t = : :t : :t = : :t : :t : :t = : :(t t ). The sides are ; ; and - there is no other one.. log = : (t t ) ln : = ln ln t t = : ln : = ln : hours ln :. a) vertical asmtote at = hole at = vertical asmtote at = ; ; hole at = ;. see handout Trigonometric Identities.. a) > and = < or > c) = + k where k Z d) = k where k Z e) = k k Z f) = + k k Z g) = + k k Z. a) c),. a) :, :, b : m :, :, b : m d) no solution c) :, b : m, : d) : : c : : : c : e) : : c : f) c = : = : = : e). a).. ft or ft.. a) r or r +... a)

10 Math Final Review - Version B age. B = ln ln A or B = A log Solution: Let f () = c d (d > ) be the eonential function eressing the oulation: Let eress the ears assed, starting at. For A, we write the equation For B, we write So A = ln ln d and B = ln ln d. Then B A = c = c d A = d A ln = A ln d ln ln d = A c = c d B = d B ln = B ln d ln ln d = B ln ln d ln ln d B A = ln ln B = A ln ln = A log = ln ln d ln d ln = ln ln Last revised: Aril,