Calculus 1st Semester Final Review

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1 Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim Find all vertical asymptotes of the graph of + f ( ) + 7 Find the limit: lim + Find the limit: lim f ( ) if f ( ) R S T + 4,, Find the vertical asymptote(s) of f ( ) + 6 At each point indicated on the graph, determine whether the value of the derivative is positive, negative, zero, or if the function has no derivative 4 Find the limit: lim + 5 Find the limit: lim 6 Let f ( ) R S T a lim f ( ) 0 b lim f ( ) 0 + c lim f ( ) 0 + +, 0 Find each limit (if it eists), > 0 7 Find the values of for which f ( ) is 4 discontinuous and label these discontinuities as removable or nonremovable 8 Let f() 5 and g() 4 a Find f(g()) b Find all values of for which f(g()) is discontinuous 4 Use the definition of a derivative to calculate the derivative of f() + 5 Find an equation of the tangent line to the graph of f() + when 6 Find the values of for all points on the graph of f() at which the slope of the tangent line is 4 7 Find all points at which the graph of f() has horizontal tangent lines 8 The position function for an object is given by s(t) 6t + 40t, where s is measured in feet and t is measured in seconds Find the velocity of the object when t seconds 9 Differentiate: y ( ) 0 Calculate d y d if y

2 Find the derivative of y + Find the derivative of y ( + + 5) 6 Find dy d for y + ( ) 6 Determine whether the Mean Value Theorem applies to f() on the interval [, ] If the Mean Value Theorem can be applied, find all value(s) of c in the interval such that f ( c) f ( ) f ( ) If the Mean Value Theorem does not apply, state why 4 The position equation for the movement of a particle is given by s (t + ) where s is measured in feet and t is measured in seconds Find the acceleration of this particle at second 7 Find the open intervals on which f ( ) decreasing is increasing or 5 Find dy if y d + y 6 Use implicit differentiation to find dy d for + y + y 5 7 Find the slope of the curve y 4 y at the point F HG, I KJ 8 Find the open intervals on which f() is increasing or decreasing 9 Use the graph to identify the open intervals on which the function is increasing or decreasing 8 The radius of a circle is increasing at the rate of 5 inches per minute At what rate is the area increasing when the radius is 0 inches? 9 Air is being pumped into a spherical balloon at a rate of 8 cubic feet per minute At what rate is the radius changing when the radius is feet? F HG 4 V πr I KJ 0 A point moves along the curve y + so that the y 0 value is decreasing at a rate of units per second Find the instantaneous rate of change of with respect to time at the point on the curve where 5 Find all critical numbers for the function: f ( ) + Find all critical numbers for the function: f() 4 4 Find the minimum and maimum values of f() + on the interval [0, ] 4 Consider f ( ) ( ) a Sketch the graph of f() b Calculate f() and f(4) c State why Rolle s Theorem does not apply to f on the interval [, 4] 5 Decide whether Rolle s Theorem can be applied to f() on the interval [, ] If Rolle s Theorem can be applied, find all value(s), c, in the interval such that f ( c) 0 If Rolle s Theorem cannot be applied, state why 40 Find all relative etrema of y ( + 4) 4 Find the relative minimum and relative maimum for f() + 4 Use the first derivative test to find the -values that give relative etrema for f() Let f ( ) Show that f has no critical numbers 44 A differentiable function f has only one critical number: Identify the relative etrema of f at (, f( )) if f ( 4) and f ( ) 45 Find the intervals on which the graph of the function f() is concave upward or downward Then find all points of inflection for the function 46 Find all points of inflection of the graph of the function f() ( 4) 47 Find all points of inflection of the graph of the function f() Let f() + Use the Second Derivative Test to determine which critical numbers, if any, give relative etrema

3 49 The graph of a polynomial function, f, is given On the same coordinate aes sketch f and f Find the horizontal asymptote for f ( ) 5 Find the limit: lim 4 5 An open bo is to be made from a square piece of material, inches on each side, by cutting equal squares from each corner and turning up the sides Find the volume of the largest bo that can be made in this manner 5 Use the techniques learned in this chapter to sketch the graph of f() A rancher has 00 feet of fencing to enclose a pasture bordered on one side by a river The river side of the pasture needs no fence Find the dimensions of the pasture that will produce a pasture with a maimum area 55 A manufacturer determines that employees on a certain production line will produce y units per month where y To obtain maimum monthly production, how many employees should be assigned to the production line? 56 The volume of a cube is claimed to be 7 cubic inches, correct to within 007 in Use differentials to estimate the propagated error in the measurement of the side of the cube

4 Calculus st Semester Final Review Reference: [66] [] The limit does not eist Reference: [] [5] y 6 Reference: [78] [] 0 Reference: [74] [] 5 Reference: [79] [4] Reference: [746] [5] Reference: [80] [6] a b c The limit does not eist Reference: [80] [7], removable;, nonremovable Reference: [8] [8] 5 a 4 b, Reference: [86] [9] 7 Reference: [98] [0] Reference: [9] [] 7 Reference: [95] [] Reference: [05] [] a no derivative b negative c zero d positive e zero Reference: [6], [6] Reference: [8] [7] (, ), (, ) Reference: [40] [8] 64 ft/sec Reference: [] [9] [0] 6 + ( ) Reference: [5] [] ( ) Reference: [] + ( + ) / Reference: [] [] ( + )( + + 5) 5 Reference: [] ( 7 + ) [] + Reference: [4] [4] 4 ft/sec Reference: [4] y [5] ( + y) + Reference: [40] y [6] + y Reference: [0] [4]

5 Reference: [49] [7] Reference: [55] [8] 00π in /min Reference: [54] 7 ft/min [9] 9π Reference: [59] [0] 5 units/sec Reference: [6], [] Reference: [64] [] 0, Reference: [67] [] Minimum at (, 0); Maimum at (, 4) Reference: [7] a Reference: [84] [8] Increasing (, 0) and (, ); decreasing (0, ) Reference: [80] [9] Decreasing (, ) and (, ) Reference: [86] [40] F HG, 54 I KJ Reference: [87], relative maimum [4] Relative maimum: (, 0); relative minimum: (, 7) Reference: [8] [4] Relative maimum at Reference: [87] [4] f ( ) 0 for all ( ) f ( ) is undefined at, a vertical asymptote Reference: [88] [44] Relative maimum Reference: [94] [45] Concave upward: (, 0), (, ) Concave downward: (0, ) Points of inflection: (0, ) and (, 4) Reference: [94] [46] (4, 0), (, ) Reference: [98] [47] (, 4) [4] b f() f(4) c f is not continuous on [, 4] Reference: [77] [5] Rolle s Theorem applies; c 0,, and Reference: [9] [48] 0, relative maimum;, relative minimum Reference: [74] [6] The Mean Value Theorem applies; c 5 Reference: [8] [7] Increasing (, 0); decreasing (0, )

6 Reference: [9] [49] Reference: [0] [50] y Reference: [07] [5] Reference: [7] [5] 8 () 8 cubic inches Reference: [8] [5] Reference: [8] [54] 75 feet by 50 feet Reference: [8] [55] 4 Reference: [4] [56] ±000 in

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