Exam. Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the given epression (), sketch the general shape of the graph of = f(). [Hint: it ma be helpful to find.] ) = ( - ) ) ) The graphs below show the first and second derivatives of a function = f(). Select a possible graph of f that passes through the point P. ) f f P P [NOTE: Graph vertical scales ma var from graph to graph.] Which of the graphs shows the solution of the given initial value problem? ) d = -, = when = - d ) (-, ) (-, ) [NOTE: Graph vertical scales ma var from graph to graph.] [NOTE: Graph vertical scales ma var from graph to graph.] (-, ) (-, ) [NOTE: Graph vertical scales ma var from graph to graph.]

2 ) Find the table that matches the graph below. ) Which of the graphs shows the solution of the given initial value problem? ) d =, = when = - d ) a cb (-, ) (-, ) f () a b f () a does not eist b f () a does not eist b does not eist f () a b c 7 c c 7 c (-, ) (-, ) ) Using the following properties of a twice-differentiable function = f(), select a possible graph of f. Derivatives < >, < - =, < - < < <, < - <, = < < <, > - =, > > >, > ) 7) Find the graph that matches the given table. f () - does not eist 7)

3 ) The graphs below show the first and second derivatives of a function = f(). Select a possible graph f that passes through the point P. ) f f P P [NOTE: Graph vertical scales ma var from graph to graph.] [NOTE: Graph vertical scales ma var from graph to graph.] [NOTE: Graph vertical scales ma var from graph to graph.] For the given epression (), sketch the general shape of the graph of = f(). [Hint: it ma be helpful to find.] 9) = -/( - ) 9) [NOTE: Graph vertical scales ma var from graph to graph.] 9 ) ) Absolute minimum onl. Absolute minimum and absolute maimum. Absolute maimum onl. No absolute etrema. ) The graphs below show the first and second derivatives of a function = f(). Select a possible graph of f that passes through point P. ) f f Determine from the graph whether the function has an absolute etreme values on the interval [a, b]. P P ) ) Absolute maimum onl. Absolute minimum and absolute maimum. Absolute minimum onl. No absolute etrema. [NOTE: Graph vertical scales ma var from graph to graph.]

4 ) Select an appropriate graph of a twice-differentiable function = f() that passes through the points (-,), -, 9, (,),, and (,), and whose first two derivatives have the 9 following sign patterns. ) : [NOTE: Graph vertical scales ma var from graph to graph.] : [NOTE: Graph vertical scales ma var from graph to graph.] [NOTE: Graph vertical scales ma var from graph to graph.] ) The graphs below show the first and second derivatives of a function = f(). Select a possible graph f that passes through the point P. ) f f P P Determine from the graph whether the function has an absolute etreme values on the interval [a, b]. ) ) [NOTE: Graph vertical scales ma var from graph to graph.] Absolute minimum and absolute maimum. Absolute maimum onl. Absolute minimum onl. No absolute etrema. ) ) [NOTE: Graph vertical scales ma var from graph to graph.] [NOTE: Graph vertical scales ma var from graph to graph.] Absolute maimum onl. Absolute minimum onl. Absolute minimum and absolute maimum. No absolute etrema. [NOTE: Graph vertical scales ma var from graph to graph.]

5 7) Find the graph that matches the given table. 7) f () [NOTE: Graph vertical scales ma var from graph to graph.] [NOTE: Graph vertical scales ma var from graph to graph.] ) The graphs below show the first and second derivatives of a function = f(). Select a possible graph f that passes through the point P. ) f f P P [NOTE: Graph vertical scales ma var from graph to graph.] [NOTE: Graph vertical scales ma var from graph to graph.] 7 Determine from the graph whether the function has an absolute etreme values on the interval [a, b]. 9) 9) ) The graphs below show the first and second derivatives of a function = f(). Select a possible graph f that passes through the point P. ) f f P P Absolute minimum onl. No absolute etrema. Absolute minimum and absolute maimum. Absolute maimum onl. ) Find the table that matches the given graph. ) a b c [NOTE: Graph vertical scales ma var from graph to graph.] f () a does not eist b c f () a does not eist b does not eist c - f () a b c - f () a does not eist b c - [NOTE: Graph vertical scales ma var from graph to graph.] 9

6 Estimate the limit b graphing the function for an appropriate domain. Confirm our estimate b using l'hopital's rule. Show each step of our calculation. ) lim «/ ) Answer the problem. 9) Use the following function and a graphing calculator to answer the questions. f() = - + +, [-.,.] 9) [NOTE: Graph vertical scales ma var from graph to graph.] a). Plot the function over the interval to see its general behavior there. Sketch the graph below [NOTE: Graph vertical scales ma var from graph to graph.] SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) Write down the first four approimations to the solution of the equation sin = using Newton's method with an initial estimate of =. ) - b). Find the interior points where f = (ou ma need to use the numerical equation solver to approimate a solution). You ma wish to plot f as well. List the points as ordered pairs (, ). Provide an appropriate response. ) Give an eample of two differentiable functions f and g with lim f() = lim g() = ««f() that satisf lim «g() = «. ) c). Find the interior points where f does not eist. List the points as ordered pairs (, ). ) For >, sketch a curve = f() that has f() = and f () = -. Can anthing be said about the concavit of such a curve? Give reasons for our answer. ) d). Evaluate the function at the endpoints and list these points as ordered pairs (, ). ) Use Newton's method to find the negative fourth root of b solving the equation - =. Start with = - and find. ) Suppose f (-) =, f () > to the right of = -, and f () < to the left of = -. Does f have a relative minimum, a relative maimum, or neither at = -? Eplain our answer. ) ) e). Find the function's absolute etreme values on the interval and identif where the occur. 7) Use Newton's method to find the four real zeros of the function f() = - + =. 7) ) If the derivative of an odd function g() is zero at = c, can anthing be said about the value of g at = -c? Give reasons for ou answer. ) Sketch a continuous curve = f() with the following properties: f() = ; f () > for > ; and f () < for <. Provide an appropriate response. ) Suppose Newton's Method is used with an initial guess o that lies at a critical point (a, b), b. What happens to and later approimations? Give reasons for our answer. Answer the question. ) Decide if the statement is true or false. If false, eplain. The points (-, -) and (, ) lie on the graph of f() =. Therefore, the Mean Value Theorem insures us that there eists some value = c on (-, ) for which f () = - (-) - (-) =. Provide an appropriate response. ) Which one is correct, and which one is wrong? Give reasons for our answers. + (a) lim - - = - lim = - + (b) lim - - = - = ) You are planning to close off a corner of the first quadrant with a line segment units long running from (,) to (,). Show that the area of the triangle enclosed b the segment is largest when =. ) ) ) ) ) ) ) Let f() = -. (a) Find f () as a piecewise defined function. Does f () eist? (b) Find all local minimum values and where the occur. (c) Find all local maimum values and where the occur. (d) Find the absolute maimum value and the absolute minimum value, or eplain wh the do not eist. 7) Use Newton's method to estimate the solutions of the equation + - =. Start with = for the right-hand solution and with = -. for the solution on the left. Then, in each case find. Provide an appropriate response. + ) Let f() = = and g() = + f'() f(). Show that lim = but lim = g'() g() =. Eplain wh this does not contradict l'hopital's Rule. 9) The function: P() = + < < «models the perimeter of a rectangle of dimensions b. (a) Find the etreme values for P. (b) Give an interpretation in terms of perimeter of the rectangle for an values found in part (a). Answer the question. ) Assume that f() and g() are two functions with the following properties: g() and f() are everwhere continuous, differentiable, and positive; f() is everwhere increasing and g() is everwhere decreasing. Which of the following functions are everwhere decreasing? Prove our assertions. i). h() = f() + g() ii). j() = f()œg() iii). k() = g() f() iv). p() = f() g() v). r() = f(g()) = (f g)() ) 7) ) 9) ) ) Use Newton's method to estimate the one real solution of the equation - - =. Start with =. Then, in each case find. )

7 Answer the problem. ) Use the following function and a graphing calculator to answer the questions. f() = +.9 sin, [, p] a). Plot the function over the interval to see its general behavior there. Sketch the graph below. - 7 b). Find the interior points where f = (ou ma need to use the numerical equation solver to approimate a solution). You ma wish to plot f as well. List the points as ordered pairs (, ). ) ) A team of engineers is testing an eperimental high-voltage fuel cell with a potential application as an emergenc back-up power suppl in cell phone transmission towers. Unfortunatel, the voltage of the prototpe cell drops with time according to the equation V(t) = -.t +.7t -.t +., where V is in volts and t is the time of operation in hours. The cell provides useful power as long as the voltage remains above <v> volts. Use Newton's method to find the useful working time of the cell to the nearest tenth of an hour (that is, solve V(t) =. volts). Use t = 7 hours as our initial guess and show all our work. ) The curve = tan crosses the line = between = and = p. Use Newton's method to find where the line and the curve cross. (Round our answer to two decimal places.) Give an appropriate answer. ) Show that the function f() = + + has eactl one zero on the interval (-«, ). ) Estimate the limit b graphing the function for an appropriate domain. Confirm our estimate b using l'hopital's rule. Show each step of our calculation. cos - 7) lim 7) e - - ) Use Newton's method to estimate the solutions of the equation + - =. Start with =. for the right-hand solution and with = - for the solution on the left. Then, in each case find. ) ) ) c). Find the interior points where f does not eist. List the points as ordered pairs (, ). d). Evaluate the function at the endpoints and list these points as ordered pairs (, ). 9) Sketch a smooth curve through the origin with the following properties: f () > for < ; f () < for > ; f () approaches as approaches -«; and f () approaches as approaches «. Answer the question. ) As moves from left to right though the point c =, is the graph of f() = + rising, or is it falling? Give reasons for our answer. 9) ) e). Find the function's absolute etreme values on the interval and identif where the occur. ) Let c() = t(p - p)p where t and p are constants. Show that c() is greatest when ) p = p. ) The graph below shows the position s = f(t) of a bod moving back and forth on a coordinate line. (a) When is the bod moving awa from the origin? Toward the origin? At approimatel what times is the (b) velocit equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration positive? Negative? ) Use Newton's method to estimate the solution of the equation sin - + =. Start with =.. Then, in each case find. Provide an appropriate response. ) Find the error in the following incorrect application of L'Hopital's Rule. lim = lim - - = - lim - = -. ) ) ) Estimate the limit b graphing the function for an appropriate domain. Confirm our estimate b using l'hopital's rule. Show each step of our calculation. ) lim sin - ) 9) Find the inflection points (if an) on the graph of the function and the coordinates of the points on the graph where the function has a local maimum or local minimum value. Then graph the function in a region large enough to show all these points simultaneousl. Add to our picture the graphs of the function's first and second derivatives. = - Estimate the limit b graphing the function for an appropriate domain. Confirm our estimate b using l'hopital's rule. Show each step of our calculation. ) lim + ) ) Let f() = - (a) Does f () eist? (b) Does f () eist? (c) Does f (-) eist? (d) Determine all etrema of f. 9) ) ) Use Newton's method to find the two real solutions of the equation =. ) ) Use Newton's method to estimate the solutions of the equation - + =. Start with =. for the right-hand solution and with = - for the solution on the left. Then, in each case find. ) Provide an appropriate response. ) Eplain wh the following four statements ask for the same information. (i) Find the roots of f() = - - (ii) Find the -coordinates of the intersections of the curve = with the line = +. (iii) Find the -coordinates of the points where the curve = - crosses the horizontal line =. (iv) Find the values of where the derivative of g() = equals zero. Answer the question. ) A marathoner ran the. mile New York Cit Marathon in. hrs. Did the runner ever eceed a speed of 9 miles per hour? Provide an appropriate response. 7) Appl Newton's method to f() = a, a >, and write an epression for n +. If the initial guess o is greater than or equal to, what happens to n + as n «? ) ) 7) ) Find the approimate values of r through r in the factorization = ( - r)( - r)( - r)( - r). ) If the derivative of an even function f() is zero at = c, can anthing be said about the value of f at = -c? Give reasons for our answer. Estimate the limit b graphing the function for an appropriate domain. Confirm our estimate b using l'hopital's rule. Show each step of our calculation ) lim ) - Give an appropriate answer. ) Show that the function r(q) = cot q + + has eactl one zero on the interval (, p). ) q ) ) 7

8 Answer the question. 7) It took seconds for the temperature to rise from e F to e F when a thermometer was taken from a freezer and placed in boiling water. Although we do not have detailed knowledge about the rate of temperature increase, we can know for certain that, at some time, the temperature was increasing at a rate of e F/sec. Eplain. Provide an appropriate response. ) Show that if h >, appling Newton's method to f() = -, - -, < 7) ) Answer the question. 7) The function: f() = - < = is zero at = and = and differentiable on (, ), but its derivative on (,) is never zero. Does this eample contradict Rolle's Theorem? 77) Use Newton's method to estimate the one real solution of =. Start with = -.and then find. 7) 77) leads to = h if = h and to = -h if = -h when < < h. 9) A manufacturer uses raw materials to produce p products each da. Suppose that each deliver of a particular material is $d, whereas the storage of that material is dollars per unit stored per da. (One unit is the amount required to produce one product). How much should be delivered ever das to minimize the average dail cost in the production ccle between deliveries? 9) Provide an appropriate response. 7) Which one is correct, and which one is wrong? Give reasons for our answers. - (a) lim - = lim = - (b) lim - = = 7) 7) The function = cot - csc has an absolute maimum value on the interval < < p. Find it. Provide an appropriate response. 7) You plan to estimate p to five decimal places b using Newton's method to solve the equation cos =. Does it matter what our starting value is? Give reasons for our answer. 7) If f() = ( - ) and g() = -, show that lim f()g() = «. 7) ( - ) 7) 7) 79) A student attempted to use l'ho^pital's Rule as follows. Identif the student's error. lim «sin (/) -- cos (/) = lim e/ «-- e/ = lim «cos (/) = e/ = ) Can anthing be said about the graph of a function = f() that has a second derivative that is alwas equal to zero? Give reasons for our answer. 79) ) Estimate the limit b graphing the function for an appropriate domain. Confirm our estimate b using l'hopital's rule. Show each step of our calculation. 7) lim 7) «7) Consider the quartic function 7) f() = a + b + c + d + e, a. Must this function have at least one critical point? Give reasons for our answer. (Hint: Must f () = for some?) How man local etreme values can f have? 7) Imagine there is a function for which f () = for all. Does such a function eist? Is it reasonable to sa that all values of are critical points for such a function? Is it reasonable to sa that all values of are etreme values for such a function. Give reasons for our answer. 7) 9 ) A square sheet of stiff paper measures in. b in., so the diagonals measure in. A pramid is to be created b removing the four congruent shaded triangles shown below, and then folding along the dotted lines. The base of the pramid is to be a square measuring in. b in. ) 7) Use Newton's method to find the positive fourth root of b solving the equation - =. Start with = and find. ) Find the inflection points (if an) on the graph of the function and the coordinates of the points on the graph where the function has a local maimum or local minimum value. Then graph the function in a region large enough to show all these points simultaneousl. Add to our picture the graphs of the function's first and second derivatives. 7) ) = - - 9) How man solutions does the equation cos =.9 - have? 9) (a) Find the height of the pramid as a function of. (b) Find the volume of the pramid as a function of. (c) Find the maimum possible volume of the pramid and the value of for which it occurs. ) Use Newton's method to estimate the one real solution of - - =. Start with = and then find. ) Let f() = (a) Find the intervals on which the function is increasing. (b) Find the intervals on which the function is decreasing. (c) Sketch a graph of = f() along with the line through (-, f(-)) and (, f()). (d) Find an values of c in the interval (-, ) that satisf f() - f(-) f (c) = - (-) Estimate the limit b graphing the function for an appropriate domain. Confirm our estimate b using l'hopital's rule. Show each step of our calculation. - cos ) lim ) Answer the question. ) Suppose that g() = - and that g (t) = - for all t. Must g(t) = -t - for all t? ) ) The position of a particle moving along the -ais is given b (t) = cos(pt) for t. ) ) ) 9) Show that g() = Provide an appropriate response. a + is an increasing function of. 9) b + (a + ) 9) Find the error in the following incorrect application of L'Hopital's Rule. sin lim + = lim cos + = lim - sin =. 9) Use Newton's method to estimate the solutions of the equation =. Start with =. for the right-hand solution and with = - for the solution on the left. Then, in each case find. 9) Give reasons for our answers. Let f() = ( - )/ (a) Does f () eist? (b) Show that the onl local etreme value of f occurs at =. (c) Does the result of (b) contradict the Etreme Value Theorem? (d) Repeat parts (a) and (b) for f() = ( - c)/. 9) Marcus Tool and Die Compan produces a specialized milling tool designed specificall for machining ceramic components. Each milling tool sells for $, so the compan's revenue in dollars for units sold is R() =. The compan's cost in dollars to produce tools can be modeled as C() = 99 + /. Use Newton's method to find the break-even point for the compan (that is, find such that C() = R()). Use = 7 as our initial guess and show all our work. 9) 9) 9) 9) (a) Give the particle's velocit as a function of t. (b) Give the particle's acceleration as a function of t. (c) For what values of t is the particle moving to the right? (d) Find the acceleration at the first instant when the particle returns to its starting position.

9 9) The accompaning figure shows a portion of the graph of a function that is twice-differentiable at all ecept at = p. At each of the labeled points, classif and as positive, negative, or zero. 9) Provide an appropriate response. ) Give an eample of two differentiable functions f and g with lim f() = lim g() = «««f() that satisf lim «g() =. ) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 9) If f() = ( - ) and g() =, show that lim ( - ) f() g() = 9) Determine the location of each local etremum of the function. ) f() = Local maimum at ; local minimum at - Local minimum at Local maimum at No local etrema ) Find the interval or intervals on which the function whose graph is shown is increasing. ) ) 97) 97) A trough is ft long as shown in the figure. Each end is a trapezoid of height ft with bottom base ft and top base ft. Water is flowing into the trough at the rate of ftper minute. Let A represent the area of the top surface of the water, let h represent the depth of the water in the trough, and let V represent the volume of the water in the trough. (a) Find epressions for A and V in terms of h. (b) Find dh dt when the depth of the water, h, is ft. Include appropriate units. (c) Find da dt when h = ft. Include appropriate units. [-, ] [-, ] (-«, -] U [, «) (-«, -] U [, «) Answer each question appropriatel. ) Find a curve = f() with the following properties: i. d d = ii. The graph passes through the point (,) and has a horizontal tangent at that point. = + = + + = + = + Sketch the graph and show all local etrema and inflection points. ) f() = /( - 7) ) ) Answer the question. 9) A trucker handed in a ticket at a toll booth showing that in hours he had covered miles on a toll road with speed limit mph. The trucker was cited for speeding. Wh? 9) 99) What can ou sa about the inflection points of the quartic curve = a + b + c + d + e, a? Give reasons for our answer. ) If f() is a differentiable function and f (c) = at an interior point c of f's domain, and if f () > for all in the domain, must f have a local minimum at = c? Eplain. 99) ) No etrema Inflection point: (,) Min: (,) No inflection points 7) The strength S of a rectangular wooden beam is proportional to its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a -in.-diameter clindrical log. (Round answers to the nearest tenth.) 7) " Local ma: -,, min:, - Inflection point: (,) Local ma: (,), min: ± 7, - 7 Inflection points: ±, w =.; d = 9. w =.; d =. w =.; d = 9. w =.; d = 7. ) How close does the semicircle = - come to the point (, )? =.7 =. =.7 =.7 Find an antiderivative of the given function. 9) 9/ / 7 /9 7 / 9 /9 Use the graph of the function f() to locate the local etrema and identif the intervals where the function is concave up and concave down. ) - - ) 9) ) Use l'hopital's Rule to evaluate the limit ) lim «+ + - «) - - Local minimum at = +; local maimum at = -; concave down on (, «); concave up on (- «, ) Local minimum at = +; local maimum at = -; concave down on (-«, «) Local minimum at = +; local maimum at = -; concave up on (, «); concave down on (- «, ) Local minimum at = +; local maimum at = -; concave up on (-«, «)

10 Find a value of a so that f is continuous at c, or indicate this is impossible. -, < ) f() = a, = ; c = - +, > - Impossible - ) Given the acceleration, initial velocit, and initial position of a bod moving along a coordinate line at time t, find the bod's position at time t. a =, v() = -, s() = s = -t + t + s = t - t + s = t - t s = t - t + Answer each question appropriatel. ) If differentiable functions = F() and = G() both solve the initial value problem d d = f(), ( ) =, on an interval I, must F() = G() for ever in I? Justif the answer. F() and G() are not unique. There are infinitel man functions that solve the initial value problem. When solving the problem there is an integration constant C that can be an value. F() and G() could each have a different constant term. F() = G() for ever in I because integrating f() results in one unique function. F() = G() for ever in I because when given an initial condition, we can find the integration constant when integrating f(). Therefore, the particular solution to the initial value problem is unique. There is not enough information given to determine if F() = G(). Solve the initial value problem. ) d = ; () =, () =, () = d = = = = - + Find an antiderivative of the given function. ) cos p + sin p sin p - cos p sin p - cos -p sin p + cos -sin p - cos ) From a thin piece of cardboard in. b in., square corners are cut out so that the sides can be folded up to make a bo. What dimensions will ield a bo of maimum volume? What is the maimum volume? Round to the nearest tenth, if necessar.. in. b. in. b. in.; 99. in..7 in. b.7 in. b.7 in.; 9. in. in. b in. b. in.; 7. in.. in. b. in. b.7 in.;,. in. ) ) ) ) ) ) 7) A compan is constructing an open-top, square-based, rectangular metal tank that will have a volume of ft. What dimensions ield the minimum surface area? Round to the nearest tenth, if necessar..7 ft b.7 ft b. ft. ft b. ft b. ft.7 ft b.7 ft b.7 ft. ft b. ft b. ft Find a value of a so that f is continuous at c, or indicate this is impossible. - + ) f() = -, a, = ; c = Impossible - Use the graph of the function f() to locate the local etrema and identif the intervals where the function is concave up and concave down. 9) Local minimum at = ; local maimum at = - ; concave up on (-«, -) and (, «); concave down on (-, ) Local minimum at = ; local maimum at = - ; concave up on (, «); concave down on (-«, ) Local minimum at = ; local maimum at = - ; concave down on (, «); concave up on (-«, ) Local minimum at = ; local maimum at = - ; concave down on (-«, -) and (, «); concave up on (-, ) Use Newton's method to estimate the requested solution of the equation. Start with given value of and then give as the estimated solution. ) - + = ; = ; Find the left-hand solution ) ) 9) ) 7 Sketch the graph and show all local etrema and inflection points. ) f() = + ) Find all possible functions with the given derivative. ) = - ) - + C + + C + + C - + C Use l'hopital's Rule to evaluate the limit ) lim - -7 ) Min: (,) No inflection point Min:, No inflection point ) The stiffness of a rectangular beam is proportional to its width times the cube of its depth. Find the dimensions of the stiffest beam than can be cut from a -in.-diameter clindrical log. (Round answers to the nearest tenth.) ).7. " w = 7.; d =. w =.; d =. w =.; d =. w = 7.; d = 9. ) Find the number of units that must be produced and sold in order to ield the maimum profit, given the following equations for revenue and cost: R() = -. C() = units 7 units units units ) Min: (,) Inflection points: -,,, Min:,- No inflection point.7 Sketch the graph and show all local etrema and inflection points. ) f() = 7 - )

11 Local minimum: -, - œ7 / œ ; local maimum: 9 Inflection point: (, ), œ7 / œ 9 Local maimum:, œ7 / œ 9 No inflection point Local maimum at, 7 No inflection points Graph the function, then find the etreme values of the function on the interval and indicate where the occur. 7) = on the interval -«< < «7) Local minimum: - Inflection point: (, ), - 7 ; local maimum:, Absolute maimum is: 9, on the interval [7, «); absolute minimum is: -9 on the interval (-«, ] Local maimum:, œ7 / œ Absolute maimum is: 9, on the interval [, «); absolute minimum is: -9 on the interval (-«,] Absolute maimum is:, on the interval [7, «); absolute minimum is: - on the interval (-«,] - Absolute maimum is:, on the interval [7, «); absolute minimum is: - on the interval (-«, ] Find the location of the indicated absolute etremum for the function. ) Minimum h() No minimum = = = - Find the etreme values of the function and where the occur. 7 9) = + The minimum value is at =. The maimum value is at = -. The minimum value is at =. The maimum value is at =. The minimum value is - 7 at = -. The maimum value is 7 at =. ) 9) Use differentiation to determine whether the integral formula is correct. ) sec - d = -7 cot - + C 7 7 No Yes )

12 Find the absolute etreme values of each function on the interval. ) f() = csc, - p p Maimum value of at = p; minimum value of - at = p Maimum value of at = -p; minimum value of - at = p Maimum value: does not eist; minimum value: does not eist Maimum value of - at = p; minimum value of at = Using the derivative of f() given below, determine the intervals on which f() is increasing or decreasing. ) f () = /( - ) Decreasing on (, ); increasing on (-«, ) U (, «) Increasing on (, «) Decreasing on (, ); increasing on (, «) Decreasing on (-«, ) U (, «); increasing on (, ) Find the location of the indicated absolute etremum for the function. ) Maimum h() ) ) ) Identif the function's etreme values in the given domain, and sa where the are assumed. Tell which of the etreme values, if an, are absolute. ) g(t) = t -.t + t, t < «) Local minimum:, Local minimum:, 7 Local minimum:, 7 Local minima: (, ) and, 7 ; Local maimum:, ; Absolute minimum:, ; Local maimum:, ; Absolute maimum:, ; Local maima: (, ) and, ; Absolute maimum:, 7 ; Local maimum:, ; Absolute minimum:, ) A private shipping compan will accept a bo for domestic shipment onl if the sum of its length and girth (distance around) does not eceed in. What dimensions will give a bo with a square end the largest possible volume? ) = No maimum = - = Use differentiation to determine whether the integral formula is correct. ) (9 + ) d = (9 + ) + C No Yes ) in. in. in. in. in. in. in. in. in. in. in. 9 in. Determine the location of each local etremum of the function. 7) f() = Local maimum at - ; local minimum at 7) Local maimum at - ; local minimum at Local maimum at -; local minimum at Local maimum at -; local minimum at. Find all possible functions with the given derivative. ) = C C C C ) Maimum: (,) No inflection point Find an antiderivative of the given function. 9) - 7 9) Answer each question appropriatel. ) The position of an object in free fall near the surface of the plane where the acceleration due to gravit has a constant magnitude of g (length-units)/sec is given b the equation: s = - gt + vt + s, where s is the height above the earth, v is the initial velocit, and s is the initial height. Give the initial value problem for this situation. Solve it to check its validit. Remember the positive direction is the upward direction. d s dt = -g d s dt = -gt, s() = s d s dt = -g, s () = v, s() = s d s dt = g, s () = v, s() = s ) - - Local minimum: (-,-) Local maimum: (,) Inflection point: (,), (-, - ),(, ) Using the derivative of f() given below, determine the critical points of f(). ) f () = ( + 7)( + ) -, -7 -, -, -7 7, ) Find the largest open interval where the function is changing as requested. ) Decreasing f() = - -, -«, «-«, -, «) - Local minimum: (,-) Local maimum: (-,) Inflection point: (,) Sketch the graph and show all local etrema and inflection points. ) f() = + ) Local minimum: (-,- ) 7

13 Local minimum: (-,- ) Local maimum: (,) Inflection point: (,) Find a value of a so that f is continuous at c, or indicate this is impossible. - sin, 7) f() = c, = 9 9 7) ) Given the velocit and initial position of a bod moving along a coordinate line at time t, find the bod's position at time t. v = cos p t, s() = ) - s = p sin p t s = p sin p t + p s = sin t s = p sin p t - ) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $ per foot for two opposite sides, and $ per foot for the other two sides. Find the dimensions of the field of area ft that would be the cheapest to enclose. 9.9 $ b.7 $ 9. $ b. $.7 $ b 9.9 $. $ b 9. $ Use a computer algebra sstem (CAS) to solve the given initial value problem. ) = ( - ), () = + = tan- - + = ln = tan- - + = tan- - ) ) 9) A small frictionless cart, attached to the wall b a spring, is pulled cm back from its rest position and released at time t = to roll back and forth for sec. Its position at time t is s = - cos pt. What is the cart's maimum speed? When is the cart moving that fast? What is the magnitude of of the acceleration then? p. cm/sec; t =. sec,. sec; acceleration is cm/sec p. cm/sec; t =. sec,. sec,. sec,. sec; acceleration is cm/sec p. cm/sec; t = sec, sec, sec, sec; acceleration is cm/sec p. cm/sec; t =. sec,. sec,. sec,. sec; acceleration is cm/sec Find the most general antiderivative. sec q ) dq sec q - cos q cos q + C cot q + C -cot q + C q + tan q + C Use l'hopital's Rule to evaluate the limit. 9) ) ) A highwa must be constructed to connect Village A with Village B. There is a rudimentar roadwa that can be upgraded mi south of the line connecting the two villages. The cost of upgrading the eisting roadwa is $, per mile, whereas the cost of constructing a new highwa is $, per mile. Find the combination of upgrading and new construction that minimizes the cost of connecting the two villages. ) ) lim p cos - - p - - ) $,, $,, $7,, $,7, ) The velocit of a particle (in ft s ) is given b v = t - t +, where t is the time (in seconds) for which it has traveled. Find the time at which the velocit is at a minimum. s s. s s Find all possible functions with the given derivative. ) = C + C + C + C ) ) 9 Find the largest open interval where the function is changing as requested. ) Increasing = ( - 9) (-«, ) (-, ) (-, ) (, «) Determine whether the function satisfies the hpotheses of the Mean Value Theorem for the given interval. ) f() = /, -, No Yes Find the most general antiderivative. ) (-7 sec ) d -7 cot + C -7 tan + C 7 cot + C tan + C 7 Use the maimum/minimum finder on a graphing calculator to determine the approimate location of all local etrema. 7) f() = Approimate local maimum at.; approimate local minima at -.9 and. Approimate local maimum at.7; approimate local minima at -.7 and.7 Approimate local maimum at.7; approimate local minima at and. Approimate local maimum at.7; approimate local minima at -. and.9 Find the derivative at each critical point and determine the local etreme values. ) = /( - ); Critical Pt. derivative Etremum Value = maimum = minimum - Critical Pt. derivative Etremum Value = Undefined local ma = minimum - Critical Pt. derivative Etremum Value = Undefined local ma = minimum - Critical Pt. derivative Etremum Value = Undefined local ma = minimum ) ) ) 7) ) Use Newton's method to estimate the requested solution of the equation. Start with given value of and then give as the estimated solution. ) = ; = ; Find the left-hand solution Find the etreme values of the function and where the occur. + ) = + + The maimum is at = ; the minimum is - at = -. None The maimum is - at = ; the minimum is at = -. The maimum is at = ; the minimum is at = -. Determine the location of each local etremum of the function. ) f() = Local maimum at -; local minimum at - Local maimum at ; local minimum at Local maimum at -; local minimum at - Local maimum at ; local minimum at Use Newton's method to estimate the requested solution of the equation. Start with given value of and then give as the estimated solution. ) - = ; = ; Find the negative solution ) + + = ; = -; Find the one real solution ) ) ) ) ) Find a value of a so that f is continuous at c, or indicate this is impossible. 9) Let f() = (sin ),. Etend the definition of f to = so that the etended function is continuous there. f() = (sin ), e, = f() = (sin ), -, = 9) Sketch the graph and show all local etrema and inflection points. ) f() = - + ) f() = (sin ),, = f() = (sin ),, = Find all possible functions with the given derivative. 7 ) = t 7t + C 7 t + C t + C 7t + C )

14 Local maimum:, Local minimum:,- Inflection point:,- No etrema Inflection point:, Find all possible functions with the given derivative. 9) = t - t 9) 7 t - t + C t - t + C t - t + C t - t + C - - Find an antiderivative of the given function. 7) csc cot 7) - cot - csc csc - csc Use l'hopital's Rule to evaluate the limit ) lim -«+ 7-7) Local ma:,, min:,- Inflection point:,- Local min:, No inflection point «9 7 L'Hopital's rule does not help with the given limit. Find the limit some other wa. 7) lim cot sin + «- 7) ) lim «+ - 7) «- - - sec 7) lim tan + 7) «- Find the etreme values of the function and where the occur. 7) = - + Local maimum at (, ), local minimum at (, -). Local minimum at (, -). None Local maimum at (, ). Use differentiation to determine whether the integral formula is correct. ) + 7 d = C Yes No 7) ) Find the most general antiderivative. 7) ( t - 7 t) dt t - t + C t / - 7 t /7 + C t / - 7 t /7 + C - t / - 7 t -/7 + C Use differentiation to determine whether the integral formula is correct. 7) ( - ) d = ( - ) + C 7) 7) Yes No Use the graph of the function f() to locate the local etrema and identif the intervals where the function is concave up and concave down. 77) 77) Find the absolute etreme values of each function on the interval. ) f() = csc, - p p Maimum value: does not eist; minimum value: does not eist Maimum value of at = p; minimum value of - at = p Maimum value of at = -p; minimum value of - at = p Maimum value of at = ; minimum value of - at = p ) Local minimum at = +.9,; local maimum at = -.9; concave down on (, «); concave up on (-«, ) Local minimum at = -.9,; local maimum at = +.9; concave up on (-«, -.9) and (.9, «); concave down on (-.9,.9) Local minimum at = +.9,; local maimum at = -.9; concave up on (-«, -.9) and (.9, «); concave down on (-.9,.9) Local minimum at = +.9,; local maimum at = -.9; concave up on (, «); concave down on (-«, ) Solve the initial value problem. dr 7) dq = - p cos p q, r() = - r = - p sin p q - r = sin p q - r = cos p q - 7 r = -sin p q - 7) Find the derivative at each critical point and determine the local etreme values. ) = -, +, > Critical Pt. derivative Etremum Value = minimum Critical Pt. derivative Etremum Value = undefined minimum Critical Pt. derivative Etremum Value = undefined minimum Critical Pt. derivative Etremum Value = minimum Use the graph of the function f() to locate the local etrema and identif the intervals where the function is concave up and concave down. ) - - ) ) 79) The acceleration of gravit near the surface of Mars is.7 m/sec. If a rock is blasted straight up from the surface with an initial velocit of m/sec (about 9 mph), how high does it go? (Hint: When is velocit zero?) Approimatel 97. meters Approimatel 9. meters Approimatel. meters Approimatel 7 meters Determine whether the function satisfies the hpotheses of the Mean Value Theorem for the given interval. ) g() = /,, No Yes 79) ) - - Local maimum at = +; local minimum at =-; concave up on (-«, «) Local minimum at = +; local maimum at =-; concave down on (, «); concave up on (-«, ) Local minimum at = +; local maimum at =-; concave up on (, «); concave down on (-«, ) Local maimum at = +; local minimum at =-; concave up on (, «); concave down on (-«, ) Find the largest open interval where the function is changing as requested. ) Decreasing f() = v - v (-«, -) (-, «) (, «) (-«, ) )

15 Determine whether the function satisfies the hpotheses of the Mean Value Theorem for the given interval. cos q q, -p q < ) f(q) =, q = ) No Yes Find a value of a so that f is continuous at c, or indicate this is impossible. -, < ) f() = a, = ; c = -, > - Impossible Use the maimum/minimum finder on a graphing calculator to determine the approimate location of all local etrema. 7) f() = Approimate local maimum at.; approimate local minimum at -.9 Approimate local maimum at.79; approimate local minima at -.7 and. Approimate local maimum at.; approimate local minima at -.9 and -.7 Approimate local maimum at.79; approimate local minimum at. Solve the initial value problem. ) d d = +, > ; () = = = + - = = - + ) 7) ) Find the function with the given derivative whose graph passes through the point P. 9) r (q) = + sec q, P(, ) r(q) = q + tan q r (q) = q + tan q r(q) = q + tan q r(q) = q + sec q 9) You are driving along a highwa at a stead 7 ft/sec when ou see a deer ahead and slam on the brakes. What constant deceleration is required to stop our car in ft?. ft/sec. ft/sec. ft/sec. ft/sec Solve the initial value problem. 9) dr dt = 9t + sec t, r(-p) = - r = 9 + tan t - r = 9t + tan t - - 9p r = 9 t + tan t p r = 9 t + cot t p 9) A rocket lifts off the surface of Earth with a constant acceleration of m/sec. How fast will the rocket be going. minutes later? 7. m/sec 7 m/sec -7 m/sec 7. m/sec Find the largest open interval where the function is changing as requested. 9) Decreasing f() = - (, «) (-, «) (-«, ) (-«, -) 9) 9) 9) 9) 9) Use l'ho^pital's rule to find the limit. 9) lim sin «9) Find all possible functions with the given derivative. 97) = C C 97) C 7 + C 9) lim «+ - - Find all possible functions with the given derivative. 9) = 7 + C 7 + C + C 7 + C 9) 9) Use the maimum/minimum finder on a graphing calculator to determine the approimate location of all local etrema. 9) f() = Approimate local maimum at.9; approimate local minima at -. and 7. Approimate local maimum at.9; approimate local minima at -.7 and 7. Approimate local maimum at.97; approimate local minima at -.9 and -.7 Approimate local maimum at -.9; approimate local minima at -.9 and 7. 99) On our moon, the acceleration of gravit is. m/sec. If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom seconds later? 7 m/sec - m/sec -7 m/sec m/sec 9) 99) 7 Sketch the graph and show all local etrema and inflection points. ) f() = + cos, p ) Local minimum: p p, - ; local maimum:, Inflection point: p, - Local minimum: (., -.); local maimum: (.,.) Inflection points: (.7,.9) and (.,.7) - Local minimum: p, p - ; local maimum: Inflection points: p, p p and, p p, p + - No local etrema. Inflection point: p, p - - 9

16 Find the location of the indicated absolute etremum for the function. ) Maimum ) Find the location of the indicated absolute etremum for the function. ) Minimum ) g() h() No maimum = = Use l'hopital's Rule to evaluate the limit. - ) lim = Find the absolute etreme values of each function on the interval. ) f() = cos - p, 7p Maimum value of at = - p ; minimum value of - at = - 9 p Maimum value of at = p ; minimum value of - at = 9 p Maimum value of at = - p ; minimum value of - at = 9 p Maimum value of at = p ; minimum value of - at = 7 p ) F() = -,. Maimum =, - ; minimum = -, - Maimum =, - ; minimum = (-, -) Maimum =, - ; minimum =, - Maimum =, ; minimum = (, -) ) ) ) = - = - = = Find the largest open interval where the function is changing as requested. ) Increasing f() = + (-«, ) (-«, ) (, «) (, «) 7) A bookstore has an annual demand for 7, copies of a best-selling book. It costs $. to store one cop for one ear, and it costs $ to place an order. Find the optimum number of copies per order.,9 copies, copies,97 copies,9 copies Use the maimum/minimum finder on a graphing calculator to determine the approimate location of all local etrema. ) f() = Approimate local maima at -. and -.; approimate local minima at -. and.7 Approimate local maima at -. and -.7; approimate local minima at -.7 and.9 Approimate local maima at -.9 and -.; approimate local minima at -. and.99 Approimate local maima at -. and -.; approimate local minima at -.7 and.7 9) A -in. piece of string is cut into two pieces. One piece is used to form a circle and the other to form a square. How should the string be cut so that the sum of the areas is a minimum? Round to the nearest tenth, if necessar. Square piece =. in., circle piece =. in. Circle piece =. in., square piece =. in. Square piece =. in., circle piece =. in. Square piece = in., circle piece = in. ) 7) ) 9) Find an antiderivative of the given function. ) ) Find the location of the indicated absolute etremum for the function. ) Minimum f() ) Find all possible functions with the given derivative. ) = csc q csc q + C - cot q + C - csc q + C - cot q + C ) ) If the price charged for a cand bar is p() cents, then thousand cand bars will be sold in a ) certain cit, where p() = 7 -. How man cand bars must be sold to maimize revenue? - cand bars thousand cand bars cand bars thousand cand bars Solve the initial value problem. ) d = -, () =, () = d = = = = Find the most general antiderivative. ) sin q(cot q + csc q) dq cos q + C sin q + q + C csc q + cos q + C sin q + C ) A trough is to be made with an end of the dimensions shown. The length of the trough is to be feet long. Onl the angle q can be varied. What value of q will maimize the trough's volume? ) ) ) - - = - = = = - Answer each question appropriatel. 7) Suppose the velocit of a bod moving along the s-ais is ds = 9.t -. dt Is it necessar to know the initial position of the bod to find the bod's displacement over some time interval? Justif our answer. Yes, knowing the initial position is the onl wa to find the eact positions at the beginning and end of the time interval. Those positions are needed to find the displacement. Yes, integration is not possible without knowing the initial position. No, displacement has nothing to do with the position of the bod. No, the initial position is necessar to find the curve s= f(t) but not necessar to find the displacement. The initial position determines the integration constant. When finding the displacement the integration constant is subtracted out. L'Hopital's rule does not help with the given limit. Find the limit some other wa. sin 7 ) lim 7) ) 7 Sketch the graph and show all local etrema and inflection points. 9) f() = + 9) q q e e

17 L'Hopital's rule does not help with the given limit. Find the limit some other wa. tan ) lim sec + ) - « ) A rectangular sheet of perimeter cm and dimensions cm b cm is to be rolled into a clinder as shown in part (a) of the figure. What values of and give the largest volume? ) Minimum: (-,-) No inflection points - Minimum: (,-) No inflection points Minimum: (,-) No inflection points Find the etreme values of the function and where the occur. ) = - None Local maimum at (, ), local minimum at (-, ). Local maimum at (-, ), local minimum at (,). Local maimum at (, -) Minimum: (-,-) No inflection points ) = cm; = cm = cm; = cm = cm; = cm = cm; = 7 cm Use the maimum/minimum finder on a graphing calculator to determine the approimate location of all local etrema. ) f() = Approimate local maimum at.7; approimate local minima at -.7 and -. Approimate local maimum at.; approimate local minima at -. and.799 Approimate local maimum at.; approimate local minima at -.9 and.7 Approimate local maimum at.; approimate local minima at -.9 and.77 ) Given the acceleration, initial velocit, and initial position of a bod moving along a coordinate line at time t, find the bod's position at time t. a = cos t, v() = -, s() = s = - sin t - t + s = sin t - t + ) ) ) A baseball team is tring to determine what price to charge for tickets. At a price of $ per ticket, it averages, people per game. For ever increase of $, it loses, people. Ever person at the game spends an average of $ on concessions. What price per ticket should be charged in order to maimize revenue? $. $. $. $7. Find all possible functions with the given derivative. ) = 9 + C + C 9 + C + C ) ) s = cos t + t + s = - cos t - t + Find the function with the given derivative whose graph passes through the point P. 7) g () = +, P(-, ) g() = g() = + - g() = -- + g() = ) Use differentiation to determine whether the integral formula is correct. ) cos d = sin + C No Yes Plot the zeros of the given polnomial on the number line together with the zeros of the first derivative. 9) = + + ) 9) ) A particle moves on a coordinate line with acceleration a = ds/dt = / t + 9 t, subject to the conditions that ds/dt = and s = when t =. Find the velocit v = ds/dt in terms of t and the position s in terms of t. v = t + t - ; s = v = t - t + ; s = 7 t + t - t + 7 t - t + t + ) v = t + t - ; s = v = 7 t + t - t + 7 t + t - t + ; s = t + t - 7 Find the location of the indicated absolute etremum for the function. ) Maimum g() 7 ) ) Given the velocit and initial position of a bod moving along a coordinate line at time t, find the bod's position at time t. v = t sin p p, s(p ) = s = - cos t p + s = - cos t p +.7 t s = - cos p +. t s = cos p + ) Find the function with the given derivative whose graph passes through the point P. ) f () = -, P(, 7) f() = - + f() = - + f() = - f() = - + ) = No maimum = - = Solve the initial value problem. ) dv dt = p csc t cot t, v = v = csc t + v = - csc t + v = - csc t + v = - sec t + ) ) Given the velocit and initial position of a bod moving along a coordinate line at time t, find the bod's position at time t. v = cos p t, s() = s = p sin p t + s = p sin p t + p s = sin t + s = p sin p t Find all possible functions with the given derivative. ) r = 7 + q ) ) r = 7q + q r = 7q - q r = 7q + q r = 7q - q 7

18 7) Given the acceleration, initial velocit, and initial position of a bod moving along a coordinate line at time t, find the bod's position at time t. a = 9., v() = -9, s() = s =.9t - 9t + s = -.9t + 9t + s =.9t - 9t s = 9.t - 9t + Find the etreme values of the function and where the occur. ) = - The maimum is at = -. The minimum is at =. The maimum is at =. The minimum is at =. 9) At noon, ship A was nautical miles due north of ship B. Ship A was sailing south at knots (nautical miles per hour; a nautical mile is ards) and continued to do so all da. Ship B was sailing east at knots and continued to do so all da. The visibilit was nautical miles. Did the ships ever sight each other? Yes. The were within nautical miles of each other. No. The closest the ever got to each other was. nautical miles. Yes. The were within nautical miles of each other. No. The closest the ever got to each other was. nautical miles. Use differentiation to determine whether the integral formula is correct. ) (9 + ) d = (9 + ) + C Yes No 7) ) 9) ) Find the etreme values of the function and where the occur. ) = + - The minimum is at =. The minimum is at = -. The minimum is - at =. The minimum is - at = -. Use the graph of the function f() to locate the local etrema and identif the intervals where the function is concave up and concave down. ) Local maimum at = ; local minimum at = - ; concave up on (, -) and (, «); concave down on (-, ) Local minimum at = ; local maimum at = - ; concave down on (, «); concave up on (-«, ) Local minimum at = ; local maimum at = - ; concave up on (, -) and (, «); concave down on (-, ) Local minimum at = ; local maimum at = - ; concave up on (, «); concave down on (-«, ) ) ) Find the absolute etreme values of each function on the interval. ) = - on [-, ] Maimum = (, ); minimum = (, -) Maimum = (, ); minimum = (, -9) Maimum = (, ); minimum = (, -7) Maimum = (, ); minimum = (-, -9) ) L'Hopital's rule does not help with the given limit. Find the limit some other wa. sec ) lim csc «- ) ) Find the optimum number of batches (to the nearest whole number) of an item that should be produced annuall (in order to minimize cost) if, units are to be made, it costs $ to store a unit for one ear, and it costs $ to set up the factor to produce each batch. batches batches batches batches ) Solve the initial value problem. ) d d = -/, () = ) = / + = / + = / - =- -7/ Find the derivative at each critical point and determine the local etreme values. 7) = - - +, , > Critical Pt. derivative Etremum Value = local ma = = 9 undefined local min undefined local ma 7 Critical Pt. derivative Etremum Value = - local min Critical Pt. derivative Etremum Value = - local ma = = - 9 undefined local min local ma 7 Critical Pt. derivative Etremum Value = - local ma 7) Find the derivative at each critical point and determine the local etreme values. ) = ( - ) Critical Pt. derivative Etremum Value = -. local ma -.77 =. local min. Critical Pt. derivative Etremum Value = -. =. local ma local min Give an appropriate answer. ) Find the value or values of c that satisf interval [-, -]. - -, Critical Pt. derivative Etremum Value =. local ma -.77 = -. local min. Critical Pt. derivative Etremum Value =. = -. local ma local min. -. f(b) - f(a) = f (c) for the function f() = + + on the b - a -, -, - ) ) = = 9 undefined local ma local min 7 = = 9 undefined local min local ma 7 L'Hopital's rule does not help with the given limit. Find the limit some other wa. ) lim «+ + 9 «) Use a computer algebra sstem (CAS) to solve the given initial value problem. ) = sin, () = = -( - ) cos + sin - 9 = - sin cos + sin + + = - cos + sin + = - cos + Identif the function's etreme values in the given domain, and sa where the are assumed. Tell which of the etreme values, if an, are absolute. 9) f() = -, -«< Local and absolute minimum: (, -9); Local and absolute maimum: (, ) Local minimum: (, -9); Local and absolute maimum: (, ) Local minimum: (, ); Local and absolute maimum: (, -9) Local and absolute minimum: (, -9); Local maimum: (, ) Use l'hopital's Rule to evaluate the limit. sin () ) f() = sin ; limit f() - ) 9) ) Plot the zeros of the given polnomial on the number line together with the zeros of the first derivative. ) = ( - )( + ) Solve the initial value problem. ) d r dt = t ; dr =, r() = dt t= r = + t + r = t + t + t r = t + t - r = -t + t ) ) 7 7