for the function f () t..4.6.8 Math : Final Eam Review Problems mallest () to largest (5) using the graph of f shown November 6, 6. Below is the graph of some function f(): 8 y om 6 4 - -4-6 4 6 8 4 6 8 4 6 8 Arrange the following numbers from smallest () to largest (5). f( + h) f() lim. h h The slope of f() at =. f(6). The average rate of change of f() from = to = 4. asing function for all. In each case, determine which dy d. =8 b) f( a ) or f( a) f( a) ) and g( ) assuming b) gis ( ) an odd function.
te D f( W) where D is measured in milligrams and 5) 5 and f (5) in terms of this pain medic rmation in part a) to estimate f (55).. Let g() = and g () =. Find g( ) and g ( ) assuming g() is an even function. g() is an odd function. h of f( ) given below to sketch a graph of f ( ).. Use the graph of f() given below to sketch a graph of f (). y f( ) 4. Determine if the following statements are true or false. If g() is continuous at = a, then g() must be differentiable at = a. the statement is true (T) or false (F). No need to mak If r () is positive then r () must be increasing. If t() is concave down, then t is ) continuous at a () must be negative. If h() has a local maimum or minimum, then at = a then gmust ( h ) be different (a) must be zero. f ( ) is positive (a) is tangent line to f() at = a. then r ( must be increasing. is ) concave 5. Sketch the down, graph of f() that then satisfies alltof ( thefollowing ) must conditions: be negative. f() is continuous and differentiable everywhere. The only solutions of f() = are =, and 4. ) has a local maimum or minimum at a then h The only solutions of f () = are = and =. The only solution of f )is the tangent line to () = is =. f( ) at a. ntaneous velocity can be positive, negative, or zero.
6. Find lim h ( + h) π π h by recognizing the definition of of f (a) for some value a. 7. Use the graph of f() below to find the values of so that lue(s) of so that f() = f () = f () = f( ) ) of so that f( ) - - - 4 5 als where here f( ) 8. Use the graph of f () below to find intervals where f() is decreasing. f() is concave down. - - - 4 5 f( ) - - - 4 5 f the following: c) y sec ( a) - - - 4 5 a f) y e ln( a) following: ) y sec ( a), f (4) 5 and f (4) 9.
9. Let a > be a constant. Find dy d y = sin(a + ) a y = a + y = sec (a) y = a + a for each of the following:. Let f() be a continuous function with f(4) =, f (4) = 5, and f (4) = 9. Find the equation of the tangent line to h() = f() + 7 at = 4. Is g() = f() Find k () where k() = f( ). increasing or decreasing at = 4? Is j() = (f()) concave up or concave down at = 4?. If g() = 6 + 5 and g (y) =, find y.. Determine where the slope of y(θ) = θ + cos (θ) will equal on the interval θ π.. Find the indicated derivatives dm dv for m = m v /c d dz ( z 9 ) 4. Find A and B so that f() is continuous and differentiable on the interval (, 5). { + 5 < f() = + A( ) + B < < 5 5. For what values of k will f() = k + k + k have an inflection point at = 5? 6. Consider the curve defined by t + 5t + 9 = 7. Find d dt. For what values of t will the tangent line be horizontal? For what values of will the tangent line be vertical? 4
7. A cable is made of of an insulating material in the shape of a long, thin cylinder of radius R. It has an electrical charge distributed evenly throughout it. The electric field, E, at a distance r from the center of the cable is given below. k is a positive constant. { kr r R E = kr r r > R. Is E continuous at r = R? Is E differentiable at r = R? Sketch E as a function of r. Find de dr. 8. Let f(t) = t + for t. Find t The critical points and determine if they are local maimum or minimum. The inflection points. The global maimum and minimum on the given interval. 9. Let f() = a + a 4 with constant a >. Find The coordinates of the local maima and minima. The coordinates of the inflection points.. Consider the family of functions f(t) = Bt. Find the values of A and B so that f(t) has + At a critical point at (4, ).. A closed rectangular bo with a square bottom has a fied volume V. It must be constructed from three different types of materials. The material used for the four sides costs $.8 for square foot; the material for the bottom costs $.9 per square foot, and the material for the top costs $.6 per square foot. Find the minimum cost for such a bo in terms of V.. An electric current, I, measured in amps, is give by I = cos(ωt) + sin(ωt) where ω is a constant. Find the maimum and minimum valus of I. For what values of t will these occur if t π. 5
ttom costs $.9 per square foot, and the material for the top e minimum cost for such a bo in terms of V. by I cos( t) sin( t) where is a values of I. For what values w of c t will these occur if in deep water is given by V( w) k c w where w is the and k are positive constants. Find the wavelength that own at the right is the he graph of the graph that will. The hypotenuse of the right triangle shown below is the segment from the origin to a point on the graph of y = 4 ( ). Find the coordinates on the graph that will maimize the area of the right triangle. is given by I cos( t) sin( t) where is a inimum values of I. For what values of t will these occur if ngle shown at the right is the oint on the graph of ates on the graph that will gle. 4. A stained glass window will be created as shown below. The cost of the semi-circular region will be $. per square foot and the cost of the rectangular region will be $8. per square foot. Due to construction constraints the outside perimeter r must be 5 feet. Find the maimum total cost of the window? What are the dimensions of the window? as shown at the right. The cost of uare foot and the cost of the ot. created Due as to shown construction at the right. The constraints, cost of the per square maimum foot and total the cost cost of the of the quare indow? foot. Due to construction constraints, et. Find the maimum total cost of the of the window? orms a triangle in the first quadrant with the -ais and b forms a triangle in the first quadrant with the -ais and area of the triangle is eactly 5. the es the triangle area of the is triangle. eactly 5. rea of the triangle. r w 5. For b >, the line b(b + )y = b forms a triangle in the first quadrant with the -ais and the y-ais. Find the value of b so that area of the triangle is eactly /5. Find the value of b so that maimizes the area of the triangle. h w h 6
M (miles) t (hours) ption in miles per gallon during the first 7 miles of the trip? During h() t, what does kt () f( ht ()) represent? Find k (.5). ). What do these quantities tell us? 6. A camera is focused on a train as the train moves along a track towards a station as shown at the right. The train travels at a constant speed of km/hr. How fast is the camera rotating (in radians/min) when the train is km from the camera? n a train as the train moves along a shown at the right. The train travels m hr. How fast is the camera when the train is km from the pile from above. It forms a right 7. Sand is poured into a pile from above. It forms a right circular cone with a base radius that radius that is always times the is always times the height of the cone. If the sand is poured at a rate of 5ft sand is being poured at a rate of h per minute, how fast is the height of the pile growing when the pile is feet high? t is the height of the pile growing 8. A voltage, V, measured in volts, applied to a resistor of R ohms produces an electrical current? of I amps where V = U R. As the currentr flows, the resistor heats up and its resistance fails. If volts is applied to a resistor of ohms, the current is initially. amps but increases by. amps per minute. At what rate is the resistance changing if the voltage 46. The rate of change of a population depends on the current population, P, and is given by remains constant? plied to a resistor dp of R ohms produces an electrical current of I amps kp ( L P ) for some positive constants k and L. rrent flows, dthe resistor 9. Theheats rate of up change and its of resistance a populationfalls. depends If on the volts current is population, P, and is given by ohms, the a) For current what is nonnegative initially. values amps of P but is increases the population dp by. increasing? amps Decreasing? For what values of P does the population remain constant? = kp (L P ) s the resistance changing if the voltage remains constant? dt d P dp b) Find for as some a function positive of P. constant For what k and value L. of P will? dt dt For what nonnegative values of P is the population increasing in time? Decreasing? For what values of P does the population remain constant? 47. The mass of a circular Find d Poil dt as slick a function of radius of P r. is For M what Kr values ln( of r P ) will, where d P K dt = is? a positive constant. What is the relationship between the rate of change of the radius. The mass a circular oil slick of radius r is M = K(r + with respect to time and the rate of change of the mass with respect to time? r + ), where K is a positive constant. What is the relationship between the rate of change of the radius with respect to time and the rate of change of the mass with respect to time? 48. A spherical snowball is melting. Its radius is decreasing at. cm per hour when the radius is. A function f(t) satisfying f (t) > has values given in the table below. 5 cm. How fast is the volume decreasing at that time? How fast is the surface area decreasing at that time?.8 Find upper and lower estimate for f(t) dt using 4 rectangles..6 49. A function f() Find t is continuous f (t)dt. and differentiable, and has values given in the table below. The. values in the table are representative of the properties of the function. a) Find upper and lower estimates for b) Find.6 f() t dt...8 f() t dt using n 4. 7 5. A function gt () is positive and decreasing everywhere. Arrange the following numbers from smallest () to largest (). t...4.6.8 f() t...4.7.
. A function g(t) is positive and decreasing everywhere. Arrange the following numbers from smallest () to largest () g(t k ) t k= 9 g(t k ) t k= lim n k= n g(t k ) t. Several objects are moving in a straight line from time t = to time t = seconds. The following are graphs of the velocities of these objects (in cm/sec). Which object is farthest from the original position at the end of seconds? Which object is closest to the original position at the end of seconds? Which object has traveled the greatest total distance during these seconds? d) Which object(s) Which object has traveled has the theleast distance total distance during these seconds? Velocity of Object A Velocity of Object B t t 4 6 8 - - - 4 6 8 Velocity of Object C Velocity of Object D t t 4 6 8 4 6 8 - - - - - 5. Illustrate the following on the graph of f( ) given below. Assume F( ) f( ). a) f() b f() a b) f() b f() a 8 b a f f
d c) d d) b b egions. Include a sketch of the regions. y(4 ) and the -ais. y and 4. Let b be a positive constant. Evaluate the following. y. (b + ) d b + d b + d ation of a particular city will grow at the rate of pt () t e per year). How many people will be added to the city in the 5. Find the eact area of the regions. Include a sketch of the regions. model? The region bounded between y = (4 ) and the -ais. The region bounded between y = + and y = +. 6. It is predicted that the population of a particular city will grow at the rate of p(t) = t + (measured in hundreds of people per year). How many people will added to the city in the first four years according to this model? ed into a tank at a constant rate of 75 gallons per hour. After e flow of water is zero according to rt () ( t) 75, gallons of water pumped into the tank. 7. At time t = water is pumped into a tank at a constant rate of 75 gallons per hour. After hours, the rate decreases until the flow of water is zero according to r(t) = (t ) + 75, gallons per hour. Find the total gallons of water pumped into the tank. 8. Use the graph of g () given below to sketch a graph of g() so that g() =. at the right to sketch a ( ) g'() g 5 5-5 4 5 6 7 8 9 - s to a stop in five seconds. Assume the deceleration is constant. velocity function. Sketch the graph of vt (). d from the time the brakes were applied until the car came to a he graph of vt () in part to aa). stop. Illustrate this quantity on the graph of v(t). position function. Sketch the graph of st (). t e dt. 9. A car going 8 ft/sec brakes to a stop in 5 seconds. Assume the deceleration is constant. Find an equation for v(t), the velocity function. Sketch the graph of v(t). Find the total distance traveled from the time the brakes were applied until the car came Find an equation for s(t), the position function. Sketch the graph of s(t). 4. Consider the following function: F () = Find F (). Find F (). ( ) c) Is F ( Is ) F () increasing or or decreasing decreasing for? for? ve down for? Is F () concave up or concave down for? 9 + 4 d. b
4. The average value of f from a to b is defined as b a b a f() d. Find the average value of f() = cos () over the interval π 4. 4. Suppose g() = f(5 + 4 cos()) and π Show that g() is an even function. Find Find π π π/ g() d. tables, g() d.. g() d = π. 4. Use the graph of f() below to determine whether the following inequalities are true or false. ln 5 4cos.. ln f() d f() d.4.5 b) Find f() d f() d ln 5 4cos( ) d.. f() d (f()) d f() d r the following. Circle True or. f() d 4 -..4.6.8.
True True - low are continuous everywhere. 5 f( ) d. n and 44. Assume all functions below are continuous everywhere. If 5 gd ( ), find n and h b5 a5 f( t 5) dt. 6f() d = 7 find 5 If g() is an odd function and If h() is an even function and If b a gd ( ). ( ) d5, find f(t) d = M, find b+5 a+5 f() d. hd ( ). g() d =, find (h() ) d = 5, find f(t 5) dt. g() d. 45. Use the graph of g () below to determine which quantity is larger. h() d. at the right to determine > gd ( ) > g ( C) > g ( B) g(c) or g(d) g (B) or g (C) g (A) or g (B) g( ) A B C D 46. Let g() = f(t) dt. In each case eplain what graphical feature of f() you used to determine the answer. t e (t) dt ln() case eplain u used to What is the sign of g(b)? What is the sign of g ( A )? What is the sign of g (A)? f( ) A B
ftdt. ( ) In each case eplain eature of f( ) you used to swer. f( ) gn of gb? ( ) A ign of g? gn of g ( A)? A 47. The graph below shows the rate, r(t), in hundreds of algae per hour, at which a population of algae is growing as a function of the number of hours. Estimate the total change in the population over the first three hours. B t the right shows the rate, rt (), in e per hour, at which a population of, where t is in hours. average value of the rate over the first total change in the population over the - -5 board that is marketed toward ineperienced ue of K so that the total area bounded by f( ) K and the -ais over the 7. following: 4 ( )d b) gtdt (). d Find 5tan(4 v) dv d. c) 5 b) Find g t dt rt 5 () tion to the initial value problems. d, s() b) d, () -5 4 ey on their first skateboard. Let Dp ( ) buy this type of skateboard when the price is p ndrical can with radius 4 inches and height 6 inches is coated with a uniform ts side (but not the top or bottom). If the ice melts at 5 cubic inches per minute, hickness of the ice decreasing when it is inch thick? 5 5 48. Find the value of K so that the total area bounded by f() = K and the -ais over the interval [, 9] is 7. the units, 49. the Suppose sign, g(t) and dt =. a practical D ( p) dp Find 5 ( ) g 5 g( t) dt. te t dt 5. A 6 foot long snake is crawling along the corner of a room. It is moving at a constant / feet per second while staying tucked snugly against the corner of the room. At the moment that the snake s head is 4 feet from the corner of the room, how fast is the distance between the head and the tail changing? r of a room. It is ing tucked snugly the snake s head is 4 stance between the t graph of a function satisfying the given all, such that lim f( ).
) lim Dp ( ) p c) 4 D( p) dp is crawling along the corner of a room. It is foot per second while staying tucked snugly room. At the moment that the snake s head is 4 he room, how fast is the distance between the ng? 5. An internet provider modeled their rate of new subscribers to an old internet service package by r(t) as shown below. Use the left hand sum rule with rectangles to estimate the total number of new subscribers between and. Rank the following from smallest to largest: 5 wing descriptions, sketch a graph of a function satisfying the given r(t) dt ose slope is increasing for all, such that lim ( ). Left hand sum with f rectangles. Right hand sum with rectangles. has a global minimum and a local maimum but not global maimum. What is the sign of r ()? Estimate r (t) dt. etermine the inflection points 4 of () modeled their rate of new ernet service package right. om smallest () to largest (): 8 New Subscribers per year 6 5 4 rt () ht if h() t ( t t)(t 5) e t. onsider the function f( ) m with n = to estimate the 4. Find a number c in the interval (,) so that scribers between and tion taneous of this estimate. rate of change of f is identical 4 to 6 the 8 average rate of change of f ove Years Since e guaranteed to find such a c? Sum with n. d Sum with n. ()? 5. Find the local linearization of π g near =. t up the integral(s) needed to find the area of the shaded region. 5. Set up the integrals needed to find the area of the shaded region below. ization of near. g f( ) g ( ). Find lim sin( ) tdt and tantdt. lim