Math 7 REVIEW Part I: Problems Using the precise definition of the it, show that [Find the that works for any arbitrarily chosen positive and show that it works] Determine the that will most likely work for any arbitrarily chosen in the proof to show that 7 8 Determine the following its: a) 6 8 ( ah) a h h c) ( ) tan d) 8 If, f( ), 8, determine f ( ), f( ), and f( ), if Let f( ), if, if a) Determine the set of all points where f will be continuous Determine the set of all points where f will be differentiable dy 6 Find : d a) y 7 y tan(sin ) 7 Using the guidelines discussed in class (domain, intercepts, symmetry, asymptotes, and sign analysis to determine intervals of increase/decrease, intervals of concavity), sketch the graph of y 9 8 If f( ), determine f ( ) and f( ) 9 Using the same guidelines as in #7 above, sketch the graph of y ( ) ( ) If y cis a tangent line to the curve The line y 6 y 7, determine c is tangent to the graph of g at Find the local etremas of f ( ) sin cos, g () Determine g() and g '() in the interval,
A particle moves on the -ais in such a way that its position at time t is given by Determine when the particle will be moving to the left s t t t t () 6 A ball is thrown vertically upward from the ground with an initial velocity of 8 ft/sec If the positive direction of the distance from the starting point is up, the equation of motion is s 6t 8 t a) Find the average velocity of the ball during the time interval t 9 Find the instantaneous velocity of the ball at seconds and at 9 seconds c) What is happening to the ball at seconds and at 9 seconds? Why? d) Find the speed of the ball at seconds and at 9 seconds e) How long will it take the ball to reach its highest point? f) Determine how high the ball will go g) How long will it take the ball to reach the ground? h) What will be the instantaneous velocity of the ball when it reaches the ground? Find ' f ( ) sin cos cos f if 6 Determine the equation of the tangent line that satisfies the following conditions a) Tangent to the graph of y sin, where, at its point of inflection Tangent to the curve y 9 y at (,) 7 Approimate, to four decimal places, using a) differentials Newton s Method 8 A closed bo with a rectangular base is needed to mail an item that requires 88 in of space Because of the unique shape of the item, the length of the base must be three times its width Find the dimensions of the bo so that the least amount of material will be used 9 An open right cylinder is inscribed in a sphere of radius meters Determine the dimensions of the cylinder that will give maimum surface area A water trough feet long has a cross section in the shape of an inverted equilateral triangle with an altitude of feet Due to a crack at the bottom of the trough, water is leaking out at the rate of ft /hour a) Determine the rate at which the height of the water is changing when the depth of the water is feet Determine the rate at which the height of the water is changing when the trough contains ft of water Evaluate: a) ( ) d If F( ) 8 t dt, find F '() 9 d Determine d d sin dt t If G G G G "( ), '() 7, and (), find ( )
Approimate 6 d, with four subintervals, using a) Trapezoidal Rule Simpson s Rule 6 Given the region bounded by y and y a) Find the area of the above region Find the volume of the solid generated when the region, where, is rotated about the y -ais 7 Find the average value of f ( ) sinon the closed interval, 8 Given the region in the first quadrant bounded by y and y a) Find the volume of the solid generated when this enclosed region is revolved about the -ais Set up, but do not integrate, an integral epression in terms of a single variable for the volume of the solid when the region is revolved about the line 9 Find the length of the arc of the curve y from (, ) to (8, ) (For #,, and ) Set up, but do not integrate, an integral that would be used to find the Volume of a solid S, where the base of S is a semi-circular disk described by y y Cross sections of S perpendicular to the -ais are semicircles with diameters on the base (, ):, Volume of the solid generated by revolving about the -ais the region bounded by y y and the line Work done in pumping all the water over the top of a right circular conical tank of altitude 8 feet and radius 6 feet The tank is full of water weighing 6 lb/ft (Answers to Part I) 8 Want to find such that whenever So we want whenever This will be true if and only if 8 Take If and, Therefore, then whenever, and so by precise minimum of, a) 8 6a c) d) f( ) 8; f( ) 8; f( ) does not eist a),,,,,, 8 n def of the it 6 a) 7 sin sec sin
7 Domain: -,--,, Intercept: (,) Symmetry with respect to origin VA at -, ; slant asymptote at y ; Crossover of slant asymptote at -,- and, Increasing on Decreasing on -,-, -,, and, Local ma at - ; loc min at ; HPI at CD on, (,); -,, PI at CU on 8 ; 9 Domain: -,--, Intercept: (,) No symmetry Hole at - No VA HA at y - ; y ; no crossovers Increasing on -,- and (-, ); no local etremas CU on -, ; CD on, ; PI at (,) c, g() 6; g'() 6 loc ma at 6, & 6, ; loc min at, &, t 6 a) y y 9 9 v( ) ft/sec; v ft/sec c) going up at t secs; going down at t secs a) ft/sec 9 d) v( ) ft/sec; v ft/sec e) sec f) s() 6 ft g) 8 sec h) -8 ft/sec 7 97 8 6 9 a) decreasing at a) ft/hr decreasing at ft/hr radius= ; height= C 6 68 6 sec a) 6 a) 8 a) 9 cu units V ( ) ( ) ( ) d 7 sq units cu units 7 V d 9 76 V d 8 6 9 8 W y y dy
Part II: Multiple Choice If at a certain point f '( ), which statement concerning the point is always correct? (A) It is either a relative maimum or a relative minimum (B) It is a point of inflection (C) It is a point at which the tangent line is parallel to the -ais (D) It is a point at which the tangent line is parallel to the y -ais (E) None of these If, for all values of, f '( ) and f "( ), which of the following curves could be part of f? A) B) C) D) E) For the function f defined in the adjacent figure, At which of the points below is f "( )? y I) A II) B III) C IV) D V) E VI) F (A) I and IV only (B) II and III only E C B A F D (C) II, III, & V only (D) VI only The radius of a sphere is increasing at the rate of inches per second When the radius is inches, its volume is increasing at the rate (in cubic inches per second) of (A) 8 (B) 8 (C) (D) The set of all number(s) c in, satisfying the conclusion of the Mean Value Theorem (for derivatives) for the function f ( ) on the interval, is (A) (B) (C) (D),
6 y - y f '( ) Let f be a function whose domain is the open interval (,) and let the derivative of f have the graph shown in the figure above The function f is decreasing on which of the following intervals? (A) (-,-) and (,) (B) (-,) and (,) (C) (-,-) and (,) (D) (-,) only (E) (,) only 7 Referring to the graph of y f '( ) in problem #6 above, f will have a point of inflection at I II III IV V (A) I, III, & V (B) II & IV (C) II only (D) III only 8 The area of the region bounded by y cos, y tan, and the y-ais can be found by (A) costand (B) cos tan (C) (tan cos d ) (D) d (tan cos d ) 9 The volume of the solid generated by revolving about the line the region bounded by y and the lines and y 8 can be found by (A) (C) ( )(8 ) 8 ( )( ) V d (B) V d (D) ( )(8 ) 8 ( )( ) V d V d The work done when a particle moves along the -ais from to under the action of a force f ( ) pounds and feet from the origin, where f( ) (), is (A) 6 ft-lb (B) 6 8 8 ft-lb (C) ft-lb (D) ft-lb ------------------------------------------------------------------------------------------------------------------------------------------- (Answers to Part II) C E C A C 6 C 7 B 8 A 9 A D 6