In #1-5, find the indicated limits. For each one, if it does not exist, tell why not. Show all necessary work.

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1 Calculus I Eam File Fall 7 Test # In #-5, find the indicated limits. For each one, if it does not eist, tell why not. Show all necessary work. lim sin.) lim.) 3.) lim cos 4.) lim 5.) lim sin 6.) Eplain, using complete sentences, reasonably good grammar and spelling, and a graph, how the derivative generalizes the idea of the slope of a line. 7.) Determine whether or not the function is continuous. Give details in your eplanation. If the function is discontinuous at any point, eplain whether the point is a removable discontinuity or a non-removable discontinuity. 3 if > f()= 4 if = 3 if > 8.) Use the δ ε definition of limit to prove that lim (3 ) = 8. 9.) Use the δ ε definition of limit to prove that lim ( ) = 6..) The graph below is the graph of f(). Answer the questions below. a.) What is lim f ( )? b.) What is lim f ( )? c.) Is f continuous at = -? Why or why not? If not, is it removable or nonremovable? d.) What is lim f ( )? 3 e.) What is lim f ( )? 3

2 f.) Is f continuous at = 3? Why or why not? If not, is it removable or nonremovable? Test # In # - 8, find dy/d. Show any necessary work. Simplify..) y = - csc.) y = (3 - ) 3.) 4.) y = ) y = sin() 6.) y = ( 4 ) (sec() - 5) 3 7.) 8.) y = 3 4 ( ) y = ( ) 4 tan 8-4 y = sin cos 9.) Let f() =. Find an equation for the tangent line to the curve when = 3..) Use the definition of derivative to find the derivative of f() = -..) Use the definition of derivative to find the derivative of f() = sin..) Use the derivative to find any points on the curve y = that have a horizontal tangent line. 3.) Suppose that the graph below is the graph of f '(). ON THE SAME SET OF AXES, draw a possible graph of f(). 4.) Suppose that the graph below is the graph of f(). ON THE SAME SET OF AXES, draw a possible graph of f '(). Test #3 In # -, find dy/d. Show any necessary work. Simplify..) y sin y = 4.) ( 4y) 6 = y d y In #3-4, find. d 3.) y = (3) 4 sec (4 ) 4.) y = y 5.) Find the 74th derivative of y = sin(3).

3 6.) A foot ladder is leaning against a wall. The bottom of the ladder starts to slide away from the wall at a rate of 5 feet per second. At the moment that the base of the ladder is 4 feet from the wall, how fast is the top of the ladder moving on the wall? 7.) Use a linear approimation to find an estimate for 8.98 /. 8.) During a sunny day a 6 foot tall man stands net to a light pole, with the light at the top of the pole. He notices that his shadow is 3 feet long while the pole's shadow is 8 feet long. a.) b.) How tall is the pole? The man comes back at night and notices his shadow that is produced by the light on the pole. He walks away from the pole at a rate of 5 feet per second. At what rate does the tip of his shadow move away from the pole? 9.) Suppose you are measuring a cone and have a possible error of ±.3 cm in any linear measure (radius, height, etc.). Given that the volume of a right circular cone is V = (/3)πr h, find the possible error in volume if the radius is measured to be 4 cm and the height 5 cm. Test #4 For #-, use the function f() = 3-5..) Use the first derivative test to find local etrema..) Use the second derivative test to find local etrema. 3.) Let f() = a.) Does Rolle's Theorem apply on the interval [-, ]? If so, find the point, c, that it guarantees. If not, why not? b.) Does the Mean Value Theorem apply on the interval [-, ]? If so, find the point, c, that it guarantees. If not, why not? In #4-5, for the given function over the given interval, determine what the Etreme Value Theorem tells us regarding the eistence of absolute etrema. If it guarantees the eistence of such etrema, find them. 4.) f() = cos, [, ] 5.) g() = , [-, ] 6.) Sketch the graph of the following function. Find the following. For any of the following, if none eist, write "NONE." 3 6 ( ) f ( ) =, f '( ) =, 3 f "( ) = ( ) ( ) ( ) 4 -intercepts, y-intercepts, relative maima, relative minima, absolute maima, intervals where: increasing, decreasing, concave up, concave down 7.) Sketch the graph of the following function. Find the following. For any of the following, if none eist, write "NONE." f() = intercepts, y-intercepts, relative maima, relative minima, absolute maima,

4 Test #5 intervals where: increasing, decreasing, concave up, concave down.) WITHOUT USING YOUR CALCULATOR AT ALL, use Newton's method to find the positive zero of f() = - 7. Start with = and do iterations. Be sure to show all work..) Use Newton's Method and your calculator to find an approimation for 3. Use = as your starting point. Do as many iterations as to get to the point where your calculator's approimations no longer change. (Be sure your mode window as "Float" chosen.) I have given more blanks than should be necessary. Once the iterations start repeating the same calculator display you may stop entering them. 3.) A farmer wants to enclose a rectangular pasture along a straight river. Additionally, he wants to put another fence line perpendicular to the river that will bisect the pasture. If the farmer has feet of fencing, what dimensions of the pasture will enclose the greatest area? Assume no fencing is necessary along the river. Be sure to make clear why we know the answer provides a maimum. 4.) Find the minimum distance between the point (, ) and the curve f() =. (Hint: If a point, (, y), is on the graph of f(), how must the and y coordinates relate?) Be sure to make clear why we know the answer provides a minimum. In #5-7, find the antiderivative of the given function. 5.) f() = 3 6.) h() = ( ) 7.) k() = sin - cos 8.) Solve the following initial value problem. f "() = - cos(), f '() = 3, f() = 9.) DO NOT USE YOUR CALCULATOR AT ALL. With n=4 and using the left hand endpoint of each interval, use rectangles to find an approimation for the area under the curve f() =, over the interval [, ]..) Do the same problem as in #9 ecept use n = 4 and use right hand endpoints. You MAY use your calculator. Test #6.) Sketch the graph of the following function. Use your calculator to find the following. 5 e y = -intercepts, y-intercepts, relative maima, relative minima, absolute maima, intervals where: increasing, decreasing, concave up, concave down..) Use Riemann sums to find 3 ( 3 ) d. Use right hand endpoints. In # - 4, find the antiderivative. All necessary work must be shown.

5 .) sin cos d.) (5 4) 3 d 3.) ( / -3/ ) d 4.) d In #5-8 evaluate the given integral. DO NOT USE YOUR CALCULATOR! 5.) cos d 6.) ππ d ) ( ) d 8.) d 3 9.) π Use your calculator to find an approimation for sin d. Write your answer to at least four decimal places..) Use the Fundamental Theorem of Calculus to find the derivative of Test #7 4 g() = (t ) dt..) Let f() = 4 3. a.) Prove algebraically that f is one-to-one. b.) Prove geometrically that f is one-to-one. c.) Find f -. In #-4, find the given limit. Show ALL necessary work..) lim sin() 3.) 5 lim 3 4 sin ( ) 4.) lim In #5-6, find the antiderivatives. ln 5.) d 6.) d / d 7.) Evaluate the definite integral. EXACT ANSWERS ONLY!! In #8-, find dy/d. 8.) y = e sin 9.) y =.) y = ln(e 3 ) 5.) a.) e e Use the fact that coth = e e to find coth -. b.) Use implicit differentiation to derive the derivative formula for y = cos -. Final Eam

6 In #-4, find the limit if it eists. Be sure to show all necessary work. 4 sin(3).) lim 5.) lim 3.) e 5 lim 4.) lim In #5-9, find the derivative of the given function. Be sure to show all necessary work ) y = 6.) y =( ) 7.) y = sec 3 tan 4 () ) y =( - )( 5) 9.) y = (sin ).) For the given function, find the absolute etrema, if any eist, on the given interval. f() = 3 3 -, [, ] In #-3, find the antiderivative of the given function. Be sure to show all necessary work..) sin cos d.) ( ) d 3.) d 4 In #4-5, evaluate the given definite integral. DO NOT USE YOUR CALCULATOR AT ALL!!! Be sure to show all necessary work. 4.) d 5.) 4 sin( π ) d 6.) Sketch the graph of the following function. Fill in the blanks below. You may use your calculator in any way it will be of help other than offering it as a bribe to get someone to do the problem for you. f ( ) = ( ) e -intercepts, y-intercepts, relative maima, relative minima, absolute maima, intervals where: increasing, decreasing, concave up, concave down. 7.) Find the point on the graph y = that is closest to the point (, -). 8.) A 4 foot ladder is leaning against a wall. The bottom of the ladder starts to slide away from the wall at a rate of 4 feet per second. At the moment that the base of the ladder is 5 feet from the wall, how fast is the top of the ladder moving on the wall? 9.) Given that y is a function of, find dy/d. y = sin(y).) Consider the following definite integral. 3 ( 3 ) d. Evaluate it using Riemann sums.