Calculus I Eam File Spring 008 Test #1 Find the following its. For each one, if it does not eist, tell why not. Show all necessary work. 1.) 4.) + 4 0 1.) 0 tan 5.) 1 1 1 1 cos 0 sin 3.) 4 16 3 1 6.) For the following function, does the Intermediate Value Theorem guarantee the eistence of a point between and 4 such that f(c) = 0? If so, find it. You may use your calculator. f() = - 3-3 7.) The origins of calculus dealt with generalizing the idea of the slope of a line to other functions. Use the idea of the secant line and the iting process to eplain how this was done. Use complete sentences and proper grammar. Draw a picture if necessary for your eplanation to make sense. 8.) The graph below is the graph of f(). Each tick mark on the aes represents one unit. Answer the questions below. a.) What is f ( )? b.) What is f ( )? + 1 1 c.) Is f continuous at = -1? Why or why not? If not, is it removable or nonremovable? d.) What is f ( )? e.) What is f ( )? + 3 3 f.) Is f continuous at = 3? Why or why not? If not, is it removable or nonremovable?
9.) Use the δ ε definition of it to prove that (3 - ) = 4. 10.) Let f() = + 1. Find an equation for the tangent line to the curve when = 3. You may not use any techniques we have not covered yet. Test # 1.) Use the definition of derivative to find the derivative of f() = 3 - + 1..) Use the definition of derivative to find the derivative of f() = sin. In #3-9, find dy/. Show any necessary work. Simplify. 3.) y = sin (cot 3 (5 6 )) 4.) - -5 y =( - ) 5.) y = (π - 3π) 6.) y = ( +3)( + 4 )( - π ) 7.) y = (3 - ) 8.) y = ( 4 + 1) (sec() - 5) 9.) y = ( + ) 4 11 tan 10.) Let f() = 3 + sin(π). Find an equation for the tangent line to the curve when =. 11.) Use the derivative to find any points on the curve y = 3 + 3-1 + 8 that have a horizontal tangent line. 1.) Find the 74th derivative of y = sin(3) + 71. 13.) If y = (3+1) 4 sec (4 ), then find Test #3 d y. 1.) Find dy/. Show all necessary work. Simplify. y sin( + y) = 4.) Find dy/. Show all necessary work. Simplify. ( + 4y) 6 = y 3.) Find d y. Do NOT leave dy/ in the answer. + y = y 4.) Use linear approimations to find an estimate for 4. 3 8. Be sure to identify the function you used, what your "nice" number is, and the linear function you construct to find the approimation. 5.) Below you see three graphs. One of them is the function, f(), one is the derivative of f, and one of them is the second derivative of f (not necessarily in that order). They relate to the position of a particle in space where is time.
Identify which of the graphs gives the position, which gives the velocity, and which gives the acceleration. Also, give reasons for your answers. Position A Velocity Acceleration Why? B C 6.) 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 feet long while the pole's shadow is 6 feet long. a.) b.) How tall is the pole? He comes back at night and notices his shadow that is produced by a light at the top of the pole. The man walks away from the pole at a rate of 4
feet per second. At what rate does the tip of his shadow move away from the pole? 7.) Water is flowing out of a tank that has the shape of an inverted cone (the small end is at the bottom). The top of the tank has a diameter of 8 feet and the tank is 10 feet tall. If water flows out of the tank at a rate of 1 cubic foot per hour, what is the rate of change of the depth when the water is 4 feet deep? (if you think it might be helpful, the formula for the volume of a cone is V = (1/3) πr h. 8.) A 4 foot long ladder rests against the side of a building. While Larry is on the ladder, his buddy, Bubba, starts backing up the truck, pushing the base of the ladder toward the building. If the base of the ladder goes toward the side of the building at a rate of 4 feet per second, how fast is the top of the ladder moving when the base is 6 feet away from the building? 9.) Suppose you are measuring a tin can and have a possible error of ±0.1 cm in any linear measure (radius, height, etc.). Given that the volume of a right circular cylinder is V = πr h, find the possible error in volume if the radius is measured to be 3 cm and the height 6 cm. Test #4 1.) Consider f() = 3 - - 4 + 1. Use the first derivative test to find local etrema. DO NOT USE YOUR CALCULATOR AT ALL!.) Consider f() = 3 - - 4 + 1. Use the second derivative test to find local etrema. DO NOT USE YOUR CALCULATOR AT ALL! 3.) Let f() = - - 3. a.) Does Rolle's Theorem apply on the interval [-1, 1]? If so, find the number, c, that it guarantees. If not, why not? b.) Does the Mean Value Theorem apply on the interval [-1, 1]? If so, find the number, 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, [0, ] 5.) g() = 3-3 - 1 + 6, [-1, ] 6.) Sketch the graph of the following function. Find the following. For any of the following, if none eist, write "NONE." 3 1 6 (1 ) f ( ) =, f '( ) =, f "( ) = ( + 1) ( + 1) 3 ( + 1) 4
-intercepts; y-intercepts; relative maima; relative minima; absolute maima; absolute minima; critical numbers; domain; possible inflection points; inflection points; range; horizontal asymptotes; vertical asymptotes; slant asymptotes; 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." Test #5 f() = 3 4-1 3 + 1 + 1 -intercepts; y-intercepts; relative maima; relative minima; absolute maima; absolute minima; critical numbers; domain; possible inflection points; inflection points; range; horizontal asymptotes; vertical asymptotes; slant asymptotes; intervals where: increasing, decreasing, concave up, concave down 1.) Use Newton's Method (doing ALL calculations by hand and showing ALL work) to find the positive root of f() = - - 1. Do only the first two iterations. Start with 0 = 1..) Use Newton's Method (using your calculator as much as you'd like) to find the positive root of f() = - - 1. Continue until the calculator display doesn't change. Start with 0 = 1. 3.) Evaluate the following antiderivative. ( + 1) 4.) Evaluate the following antiderivative. sec 5.) Evaluate the following antiderivative. cot 6.) Evaluate the following antiderivative, with the given initial conditions. y " = + 3 +, y '(0) =, y(0) = 3 7.) You have a fifteen inch long piece of wire. You are to bend it into the following shape. The triangle is to be an equilateral triangle (all side lengths the same). Find the lengths, and y, that make the enclosed area a maimum.
8.) Find the minimum distance between the point (, 0) and the curve f() = + 1. (Hint: If a point, (, y), is on the graph of f(), how must the and y coordinates relate?) 9.) Use right sums and n = 4 (and doing ALL calculations by hand and showing ALL work) to use rectangles for finding an approimation for 3. 1 + 10.) Use right sums and n = 90 (and using your calculator - be sure to show what you enter into your calculator) to use rectangles for finding an approimation for 3. 1 + 11.) Use right sums and Riemann sums to find 3. 1 + Test #6 In #1-5, find dy/. Show all necessary work. Simplify. 1.) y = 3 + 3 - - cos.) y = (sin ) sin 3.) 3 sin e + 1 y = 4.) y = ln(e + ) 5 ( + 1) 14 5.) y = (t +1 ) 0 dt. 6.) Find the antiderivative. ln 7.) π Use your calculator to find an approimation for 0 sin. Write your answer to at least four decimal places. In #8-11, evaluate the definite integral. NO CALCULATORS!! ( ) / 8.) + 1 9.). 1 cos ππ / + 1 10.) 11.) 1 3 1 + 6 + 3
1.) Suppose f is a one-to-one function such that f(3) = 8 and f ' (3) = 1/4. For each of the following, fill in the correct value or write "X" if insufficient information is given to answer the question. f -1 (3) = f -1 (8) = f -1 (1/4) = (f -1 )'(3) = (f -1 )'(8) = (f -1 )'(1/4) = 13.) Let f() = 3 + 1. a.) Prove algebraically that f is one-to-one. b.) Prove geometrically that f is one-to-one. c.) Find f -1. 14.) Showing all work, find the following antiderivative. + e ln Test #7 In #1-4, find dy/. 1.) y = sec -1 (e ).) y = 3 + sin(3) 3.) y = tan -1 ( + 3) 4.) y = log 7 ( + cos ) 5.) Use implicit differentiation to derive the derivative formula for y = sin -1. 6.) Find the antiderivative e 1 e In #7-8, evaluate the given definite integral. EXACT ANSWERS ONLY, SO NO CALCULATOR!!! 1 1 7.). 8.) 0 4 1 + 1 + 3 9.) Use your calculator to evaluate the definite integral 1. 0 e 10.) The rate of growth of a population of bacteria is proportional to the population at any given time. At noon there were 80 bacteria. At p.m. the same day there were 34. Starting with the differential equation implied by the first sentence, find how many there are at 4 p.m. the same day. In #11 - #13, find the it. If it does not eist, tell why not. You must show all necessary work.
11.) sin( t ) t 0 t tan t 1.) ( sin ) 0 + 13.) 1 14.) Find the following antiderivative. + + Final Eam e For #1-4, evaluate the given it. All necessary work must be shown. EXACT ANSWERS ONLY!!!!!!!!! 1.) +.) 3.) 0 + 5 0 + 3 + 1 sin ( t t tan t 0 ) t 1 4.) sin 0 In #5-9, find dy/. Show any necessary work. Simplify. - 1 5 5.) y = 6.) y = () 7.) - 3 5 y 8.) y = sec ( ) 9.) y = 10.) Find y ". y =( - )( + 5) 3 + 3+ 1 sin ( ) y = e + ln In #11-13, find the antiderivatives. 11.) cosh e 1.) 1 + sinh 13.) 4 ( + 1) 1 e 14.) Use the definition of tanh to find its derivative. + 1 + 3 In #15-17, evaluate the definite integral. EXACT ANSWERS ONLY!! 15.) 1 16.) 0 17.) e 1 + 3 1 + 3 18.) Show that of all the rectangles with a given area, call it A, the square has the smallest perimeter. 19.) Water is flowing out of a tank that has the shape of an inverted cone (the small end is at the bottom). The top of the tank has a radius of 6 feet and the tank is 10 feet tall. If the volume of the water is decreasing at a rate of 1 cubic foot per hour, what is the rate of change of the depth when the water is 4 feet deep? (if you think it might be helpful, the formula for the volume of a cone is V = (1/3) πr h. 0.) Sketch the graph of the function, f() = sin, 0 π. Using your calculator as much as you can, find the following relative maima, absolute minima, critical numbers, -intercepts, intervals where concave up, intervals where concave down, inflection points, range