Calculus First Exam (50 Minutes)

Transcription

1 Calculus - First Eam 50 Minutes) Friday October I Find the it or show that it does not eist. Justify your answer. t 3 ) t t t 3 ) t 3 t t 3 3) t t t 3 t t t = t t = t t 3 t 3 t = 33 3 = 6 8 t 3 t t = t )t + t + ) t + t + ) = = 3 t t )t + ) t t + ) II Find the it or show that it does not eist. Justify your answer. sin3) ) Since sin3) for all we have sin3) sin3) for > 0. Hence = 0 by the Squeeze theorem. sin ) ) 0 sin ) sin ) sin ) sinu) = = = = 0 = u 0 u 0 3) π/3 sin) sinπ/3)) π/3

2 Let f) = sin) and a = π/3. Then f ) = cos) and sin) sinπ/3) f) fa)) = = f a) = cosπ/3) = π/3 π/3 a a. III ) Find f ) and f ) if f) = sin 3 ). f ) = cos 3 ) ) 3, f ) = sin 3 ) ) 3 ) + cos 3 ) ) 6. ) Find g 3) if h 9) = 7 and g) = h ). g ) = h ), g 3) = h 9)6 = 0. IV Find the constant c which makes g continuous on, ), g) = { if < 4, c + 0 if 4. By the it laws, g) is continuous at any 4. g) = 4 4 = 6, g) = c + 0 = 4c The function g) is continuous when these are equal, i.e. when c =. V Consider the curve y + y =. ) Find the equation of the tangent line to the curve at the point P, 3). Differentiate: y dy d + y + dy = 0. d Evaluate at the point P, 3): 6 dy dy 4 = 0. d d,y)=,3),y)=,3) Solve: dy d,y)=,3) = 8.

3 This is the slope of the tangent line. The point P, 3) lies on the tangent line so the equation of the tangent line is y 3) = ) Find d y at the point P, 3). d ). 8 Differentiate again: [ ) dy ] [ ] + y d y + dy dy d d d + d + d y = 0. d Evaluate at, y) =, 3): [ ) + 3 d y 8 d,y)=,3) ] [ 8 + d y d,y)=,3) ] = 0. Solve: d y d,y)=,3) = ) VI Consider the function y = f) whose graph is shown below. Match the epression in the left column with the correct corresponding value in the right column. f 0) 6 f 0.9) 0 f ) 3 f.73) 0.6

4 For positive, the slope of the tangent line is decreasing as increases, so f 0) > f 0.9) > f ) > f.73). The only possibility is f 0) = 3, f 0.9) = 0.6, f ) = 0, f.73) = 6. VII Prove the product rule fg) = f g + fg using the definition of the derivative, high school algebra, and the appropriate it laws. Justify each step. Let w) = f)g) and assume that f and g are differentiable. Reasons: w w + h) w) ) = h 0 h ) Definition of w ). ) Definition of w). 3) High school algebra. = f + h)g + h) f)g) h 0 h ) = f + h) f) g + h) g) g + h) + f) h 0 h h 3) = f + h) f) g + h) g) g + h) + f) h 0 h h 0 h 0 h 4) = f )g) + f)g ). 5) )

5 4) Limit laws. 5) Definition of f ), continuity of g), and definition of g ). In step 5) we used the fact that a differentiable function is continuous. Here is the proof: ) g + h) g) = g + h) g)) h 0 h 0 = h 0 so h 0 g + h) = g), i.e. g is continuous. g + h) g) h h g + h) g) = h h 0 h h 0 = g ) 0 = 0. Calculus - Second Eam Two Hours) Thursday October I a) Find f ) if f) = e5 + e. f ) = 5e5 + e ) e 5 e + e ). b) Find dθ dt when θ = sin t ). Let y = t so θ = sin y). Then dθ dt = dθ dy dy dt = t = t. y t 4 II After 3 days a sample of radon- decayed to 58% of its original amount. a) What is the half life of radon-? Let Y be the amount of radon- at time t and Y 0 be the amount when t = 0. Thus Y = Y 0e ct for all time t. When t = 3 we have Y = 0.58Y 0. Hence 0.58Y 0 = Y 0e c3. Apply ln to find c: c = ln0.58). The half life is the time τ when 3 Y = 0.5Y 0 i.e. 0.5Y 0 = Y 0e cτ. Apply ln to find τ: τ = ln0.5) c = 3 ln0.5) ln0.58). b) How long would it take for the sample to decrease to 0% of its original amount?

6 The desired time t satisfies 0.0Y 0 = Y 0e ct. To find t cancel Y + 0 and apply ln : t = ln0.0) = 3 ln0.0) c ln0.58). III a) Find the inverse function gy) to the function f) = 9 +. For y 0 we have y = 9 + y = 9 + = y 9. Thus gy) = f y) = y 9. b) What is the domain of f)? When a function is given by a formula, the domain is the set of for which the formula is meaningful, unless the contrary is asserted. The domain of f is 9. c) What is the domain of gy)? The formula defining gy) is meaningful for all y, but g is defined to be the inverse function to f. Hence domaing) = rangef). Thus domaing) is the set of all y 0. IV You are videotaping a race from a stand 3 feet from the track, following a car that is moving at a constant velocity along a straight track. When the car is directly in front of you, the camera angle θ is changing at a rate of dθ = 6 radians per second. dt θ=0 43 θ 3 car a) How fast is the car going?

7 Choose coordinates so that the car moves along the -ais and the viewing stand is on the y-ais at the point Q0, 3). Let P, 0) be the position of the car. Let t be the time in seconds and suppose t = 0 is the time when the car is directly in front of you. Let θ be the angle between the y-ais and QP. We are given that Since we have d dt = v is constant, and dθ so evaluating at t = 0 gives: v = 3 dθ dt = 3 tan θ dt t=0 v = d dt = 3 sec θ) dθ dt t=0 = = b) How fast will the camera angle θ be changing a half second later? The position of the car at time t is = vt since v is constant. Thus ) θ = tan vt. 3 By the chain rule: ) dθ dt = v + ) vt. 3 3 Evaluate at t = / and plug in the value of the constant v which we found above: ) dθ v = dt t=/ + 6 ) v = 3 + ) 3. 43) 3 43 V Find sin sin3)) eactly. Justify your reasoning. For π θ π we have θ = sin sin3)) sinθ) = sin3). Also sinπ 3) = sin3) and π π 3 π. Therefore sin sin3)) = π 3. The horizontal line y = sin3) intersects the curve y = sintheta) infinitely often.)

8 VI a) Find the equation for the tangent line to y = ln) at the point =. The equation for the tangent line is y = L) where L) is the linear approimation of y = ln) at =. Using the formula L) = fa) + f a) a) with f) = ln and a = and f ) = / we get f) = ln) = 0, f ) = and so L) = 0 + ) = ). b) Find the quadratic approimation Q) to the function f) = ln at. Using the formula Q) = fa) + f a) ) + f a) a) with f) = ln). we get f ) = /, f ) = / so f ) = and hence Q) = ) ). c) Estimate ln.) without a calculator. ln.) Q.) = 0. 0.) / = VII Find n Using the formula + n) n. Justify your steps. e = + m ) m m) implies e = + n/ = + n n) n ) /. n n) so squaring gives: + ) n = e n n The wording of the question leaves some doubt as to whether the professor wants a proof of ). To be on the safe side we ll provide it: ln + ) m ) = m m ln + ) m ln + ) ln) = m ln + h) ln) =. m m m /m h 0 h

9 The it on the right is the derivative of the ln) evaluated at =. ln ) = / = so ln + ) m = so m m + m m = e. You can also use l Hospital s rule. This is VIII Suppose g is the inverse function to f) = Find g ) and g 3) = gy) y = f). Since f0) = we have g) = 0. Since f) = 3 we have g3) =. Now gf)) = so by the chain rule When = 0 this gives g f)) = f ) = When = this gives g ) = g f0)) = g 3) = g f)) = = = 6. IX a) If f) = ln what is f )? = e ln so ln = e ln ). Hence by the Chain Rule f ) = d ) eln = e ln ) d d d ln ) ln ) d = e ln ) d ln = eln ) ln ). b) Epress du in terms of t and u if u = t lnt + u). dt Use implicit dfifferentiation: Solve for du/dt: so du dt = lnt + u) + t t ) du t + u dt du dt = t t + u t + u = lnt + u) + t + du dt ) lnt + u) + ). t + u ; t t + u ).

10 X a) Find sin t t 3 t. t 0 By l Hospital s Rule: t 0 sin t t 3 t = t 0 b) Find ln + ) ln). ) cos t 3t =. + ln + ) ln) = ln = ln + ). Hence ln + ) ln) = ln + ) = ln). Calculus - Third Eam 50 Minutes) Friday December I The second derivative of a function f) satisfies f ) = 4. Moreover, f 0) = 0 and f) = 0. a) Find the function f). f ) = C so 0 = f 0) = C so f ) = f) = 4 + C so 0 = f) = 4 + C so f) = 4 +. b) Draw a graph of f). Indicate all asymptotes if any), local maima and minima, inflection points, intervals where f is increasing, intervals where f is concave up like a cup. f) f ) f ) no hor asymp < +, CU local min < < / 3 + +, CU / inflection / 3 < < 0 +, CD 0 0 local ma 0 < < / 3, CD / 3 0 inflection / 3 < < +, CU local min < + +, CU no hor asymp

11 .4 f) II a) Evaluate 4 b) Evaluate d. d = 3 )3/ 4 cos/t) dt. 3 = 3 33/ 3 3/ t Let u = /t so du = dt/t and t = = u = /, t = 3 = u = /3. Thus 3 /3 cos/t) dt = cos u du = sin/3) + sin/). t / III a) Evaluate d d If F t) = sint 4 ) then d sint 4 ) dt = d d d b) Evaluate sint 4 ) dt. F ) F ) d dt sint4 ) dt ) = F ) F ) = sin 4 ) sin 8 ).

12 If F t) = sint 4 ) then d dt sint4 ) = F t) so d dt sint4 ) dt = F ) F ) = sin 4 ) sin 8 ). 3+h IV Evaluate 3 h 0 h d. 3 The key step in the proof of the Fundamental Theorem is Hence h 0 h a+h h 0 a 3+h 3 f) d = fa). 3 d = V a) The numbers a = 0 < < < < n = b divide the closed interval [a, b] into n tiny intervals of equal length. Give a formula for i, the right endpoint of the i-th interval. i = a + b) Evaluate ib a). Note that i = i i = b a n n n n i= 5 + 3i n ) 7 3. Hint: This is a Riemann sum n where i = i. If a = 5 and b = 8 then i = 5 + 3i and n i = 3/n. If f) = 7 then n 8 8 n i= 5 + 3i n ) 7 b 3 n = f) d = a 5 7 d = = VI An athlete is at point A on a bank of a straight river, 3 km wide and wants to reach B, 8 km downstream on the opposite bank, as quickly as possible. He will row to a point X, km downstream and on the side opposite A and then run along the shore to B. If he rows at 6 km/hour and runs at 8 km/hour, what should be? See Tet Page 97 Eample 4.

13 VII Find positive numbers A and B such that 0 < A 3 + d B The graph of f) = + is shown below. There is no best answer here. What counts is the reasoning. The answers A = and B = are correct provided that you can prove the above inequality. 5 4 f) A crucial property of integration is that b a f) d g) for a b. Since + 0 for 3. We have = 3 d 3 + d 3 b a 0 d = 0. g) d if f) Calculus - Final Eam Two Hours) Monday December I In each of the following, find dy/d. a) y = sin. dy = cos. d

14 b) y = sin ). dy d = cos sin ). c) y = sin ). dy d = cos ) ) ) d) y = sin ). dy d =. II Evaluate the integral. cos d a) sin. b) c) sin ) d. 3 d III Evaluate the it. 3 a). b) 0 3 sin sin. c) n n i= cos 3 + 4i ) 4 n n IV a) Find an equation for the tangent line to the curve y + cosy) = π 3 at the point, y) = 3, π). y π = 3 π. b) A function f) satisfies the identity f) + cosf)) = π 3 and moreover f3) = π. Find the linearization L) of f) at 3. Hint: Note the similarity to part a).

15 L) = π + 3 π. V a) Find the point on the curve y =, 0 4 which is closest to, 0). b) Find the point on the curve y =, 0 4 which is farthest from, 0). VI The count in a bacteria culture was 400 after hours and 5,600 after 8 hours. a) Give a formula for the count after t hours. The general formula is N = N 0e kt. We are given t = = N = 400 and t = 8 = N = 5, 600; i.e. 400 = N 0e k, 5, 600 = N 0e 8k. Dividing gives 64 = e 6k so as 6 = 64) e k = and N 0 = 400/ = 00. Thus N = 00 t. b) How long does it take for the count to double? From N = 00 t it follows that the count doubles every hour. VII Let f) = a) Find the absolute maimum and the absolute minimum of f) on the interval 4, The derivative f ) = ) 3 ) 6 3 = ) ) vanishes at =. Moreover f) = /7, f) = /, and f4) = /0. Therefore the maimum value is / and the minimum is /0., b) Find positive numbers A and B such that 0 < A 4 d B.

16 You must justify your answer. Therefore If m f) M for a b then mb a) b a f) d Mb a) ) d ). 3 VIII A particle moves along a straight line so that its velocity at time t is v = ds dt = t3 4t = t 4)t where t is the time in minutes. a) Write an integral which gives the displacement the net change in position) during the first five minutes. b) Write an integral which gives the total distance travelled during the first five minutes. IX A tank is constructed by rotating the area bounded by the y-ais, the line y = π/ and the curve = sin y about the y-ais. Note: Not about the -ais.) Assume that the unit of length is feet = sin y

17 a) Write a definite integral which gives the volume of this tank. You need not evaluate the integral. b) The tank is being filled with water at the rate of 7 cubic feet per minute. How fast is the water level h increasing when h = π/3 feet? Hint: What is the volume of the water when the water level is h? Calculus Quiz Monday Sep 5, Minutes) z ) = a y a) z ay b) yz a c) az y d) ayz e) None of the above. z z ) = b) a y a y ) y y = zy a. a + 3) 8 a + 3) = a) a + 3) 4 b) a + ) 6 c) a d) a e) None of the above. e) a + 3) 8 a + 3) = a + 3)6. w 3 6t w = a) w 6t 3 w b) w 6t 3w c) w t) 3w d) w 8t e) None of the above.

18 e) w 3 6t w = w 3w 8t 3w = w 8t. 3w 4 = a) 0 b) c) d) e) None of the above. d) = ) + ) = + =. 5 8vw 3v w ) 3 = a) 6 v 3 b) 6 v 4 w 3 c) 3v 4 w 3 d) 3v 5 w 4 e) None of the above. d) 8vw 3v w ) 3 = 8vw 7v 6 w 6 = 3v 5 w a + 6 = a) a + b) a + 4 c) a + 4 d) a 0 e) None of the above. b) 4a + 6 = 4a + 4) = 4 a + 4 = a + 4. The following two questions are related =

19 a) b) + c) d) 4) 4) e) None of the above. b) = 4 = 4 + ) ) = 4 ) = = a) b) c) d) Does not eist e) None of the above. c) = = + =. The following two questions are related. 9 a) a ) = a) 0 b) 4 c) a a a) d) a e) None of the above. d) a). a a) a ) = a) a ) a a)a ) = = a a 0 a a) a ) = a) 0 b) 4 c) Does not eist d) e) None of the above.

20 e). a a) a a) a a) ) = = = a a a a a = ) = a) 0 b) + c) d) e) None of the above. d) + =. ) + = ) = = = a) 0 b) / c) d) Does not eist e) None of the above. e) = = 5. Read carefully. The following two questions are different. 3 h 0 a) + h h b) = c) d) 0 e) None of the above. b) h 0 h 0 + h + =. + h h = h 0 + h ) + h + ) h + h + ) = h 0 + h) h + h + ) =

21 4 h a) + h h b) = c) d) 0 e) None of the above. d) h + h h = h h + h h = = 0. ) 5 cos = a) 0 b) c) d) Does not eist e) None of the above. b) cos ) = cos ) = cos0) = cos 6 = a) 0 b) c) d) Does not eist e) None of the above. a) for > 0. cos = 0 by the Squeeze Theorem as cos Calculus Eam Tuesday October 7, 997 I Find dy d and d y d if y = +.

22 II 7 What is? III Find f t) and f t) if ft) = + t p ) q. Here p and q are constants.) IV A function y = f) is implicitly given by the equation y 3 + y = Note that the points, y) =, ), 0, 0),, ) lie on the graph. Complete the following table: 0 f) f ) f ) V A tank holds 600 cubic feet of water which drains from the bottom in 0 minutes. According to Torricelli s Law, the volume of water remaining in the tank after t minutes is V = 600 t ). 0 a) How fast is the volume decreasing at time t =? b) The tank is a cylinder whose base has area 00 square feet. Find the water level h and its rate of decrease at time t =.

23 VI The functions f) and g) are differentiable and the following table gives some of the values of f) and g) and their derivatives. Use these data to answer questions a) - e) below. If there is not enough information given to determine some answer, write INSUFFICIENT DATA f) 4 7 f ) g) g ) 5 3 a) Find fg + )) when =. b) Find f + ) when =. c) Find d) Find d d f + ). = d d fg)). = d) Find the equation for the tangent line to y = f) at the point P, f)). VII The entire length of the underside of the hour hand of a clock is coated with a thick blue dye which rubs off on the face of the clock leaving a circular blue sector. The length of the hour hand is 7 inches. The minute hand of the clock has been removed. The clock is started a noon with a clean face. a) How fast is the area of blue sector increasing two hours later?

24 φ b) How fast is the distance from the tip to the top increasing two hours later? tip means the end of the hour hand, top is the position of the tip at noon. Both tip and top are always 7 inches from the center of the clock.) VIII Prove that the derivative of the sine function is the cosine function. In your proof you may use the Limit Laws and high school algebra without further justification. You need not prove that the sine and cosine functions are continuous. You may use without proof the formulas sinh) cosh) =, = 0. h 0 h h 0 h Calculus Eam Thursday October 3, 997 I ) Find dy d and d y d if y = e. ) Find f ) if f) = log 5 ) II Evaluate each it. Simplify your answers; i.e. do not leave inverse trig functions in the final form.

25 ) cos ) + ) tan 4 ) e u + e 4u 3) u e u + e 3u III Find the quadratic approimation to f) = ln) near and use it to approimate the number ln.). IV If g is the inverse function to f) = + + e, find g ). V Radioactive strontium has a half life of 5 years. ) Give a formula for the mass of strontium that remains from a sample of 8 grams after t years. ) How long would it take for the mass to decay to grams? 3) What is the rate of decay in grams per year) when there are grams left? VI Let C) = e 3 + e 3 and S) = e 3 e 3. ) Epress S ) in tems of C). ) Draw the graph of y = C) S). VII Let f) = arctan). State and derive the formula for f ). You may assume without proof the formula for the derivative of the tangent function. Note: arctan) = tan ) tan ).)

26 VIII An etraterrestial bug swarm occupies a spherical ball in orbit about the planet Venus. The number of bugs grows eponentially at a rate of 9% per hour, i.e. the growth constant is k = The volume of the spherical bug ball is proportional to the number of bugs in the ball, the density of bugs being c = 0 6 bugs per cubic meter. There are initially N 0 = 0 5 bugs present. ) Write a formula for the number N of bugs present after t hours. ) Write formulas for the volume V, the radius R, and the surface area A of the bug sphere after t hours. 3) How fast is the surface area of the bug sphere increasing after 3 hours? Calculus Eam Thursday November 0, 997 I Graph f) = 3 for > 0. Indicate all horizontal and 3 vertical asymptotes, critical points, and points of inflection. II Find an equation for the line through the point 3, 5) that cuts off the least area from the first quadrant. 3, 5)

27 III a) Evaluate as an area. b) Evaluate / d by interpreting the definite integral d by interpreting the definite integral as an area. Hint: Sketch y = and write the area in question as the area of a sector plus the area of a triangle. IV State the Mean Value Theorem and use it to prove that a function whose derivative is positive on an interval is increasing on that interval. V a) Show that and an eplanation.) 3 b) Find a number A > /3 such that A answer.) d ). Give a picture 5 3 d. Justify your VI a) Find 9 3f) 4g)) d given that 7 f) d = 3, b) Epress the it 9 7 n n k= f) d = 4, 9 g) d =, 3 n ln 5 + 3k ) as a definite integral. n

28 n 3 c) Is the finite sum k= n ln 5 + 3k ) n integral in part b)? Give reason.) bigger or smaller than the VII Find f) if f ) = 3 and the absolute minimum value of f) is 0. Calculus Quiz Wednesday December 0, Minutes) I a) Evaluate F ) = b) Find F ). 0 t dt + t. II Find the volume of the solid obtained by rotating about the y-ais the region bounded on the left by the line = 3, on the right by the line = 4, below by the line y =, and above by the curve y = 5 ) /3. III b) Find a) Find d d t 3 t 3 sint 4 ) dt. d d sin4 ) d.

29 Calculus Final Eam Monday December 5, 997 I a) Find dy d when y = ln + e ). b) Find d y for y as in a). d II a) Evaluate 3 d 3. ) sint) b) Evaluate dt t III A population of bacteria triples in three hours. Assuming eponential growth, how long does it take to double? IV Find the points on the hyperbola y = 4 which are closest to the point, 0). V a) Find the equation for the tangent line to the curve y = at the point, y) =, ). b) Is this tangent line above the curve? Why or why not? [ n VI a) Evaluate n 3 i= + i n ) 5 ] 6 n.

30 b) Evaluate tan ). VII The velocity function for a particle moving along a line is vt) = 3t, 0 t 5. a) Find the displacement net change in position) from t = 0 to t = 5. b) Find the total distance travelled from t = 0 to t = 5. Note that the velocity changes sign.) VIII Find the interval on which the curve y = concave up. 0 dt + t + t is IX Find the volume generated by revolving the region bounded by the lines = a, = b, y = 0 and the curve y = about the -ais. Assume that a and b are constants and 0 < a < b <.) = a = b =

31 X State and prove the formula for the derivative of the inverse sine function sin ). You may assume without proof that the derivative of the sine function is the cosine function. EXTRA CREDIT A high speed train accelerates at /) meter/sec until it reaches its maimum cruising speed of 30 meters per second; after it reaches this maimum cruising speed it remains at that speed. If it starts from rest, how far will it go in 0 minutes? Calculus - First Eam Tuesday October 6, 998 I Find the it. If the it does not eist, write DNE. Distinguish between a it which is infinite and one which does not eist.) a) = 9 b) 3 3 = ) c) cos = cos d) = II Suppose that f) = 5. a) Find the first derivative f ). b) Find the second derivative f ).

32 c) Write an equation for the tangent line to the curve y = f) at the point 3, 4). III Suppose that u) and v) are differentiable. Let f) = u) v) Prove the quotient rule: f ) = u )v) u)v ) v) IV Find the it. If the it does not eist, write DNE. Distinguish between a it which is infinite and one which does not eist.) a) π tan) = b) π/ tan) = c) tan) = d) π/ tan) = e) π/4 f) π/ tan) + ) tan) tan) = Hint: Below is part of the graph of y = tan). =

33 V Find an equation for the tangent line to the curve y 5 +y = 0 at the point, ). VI A ladder 0 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at the rate of 7 feet per minute, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall? VII f). a) State the definition of the derivative f ) of a function b) 5) 4/3 =? Circle one.) none of these c) Use your knowledge of differentiation to estimate the number 7) 4/3 5) 4/3.

34 approimately without a calculator. Your answer should have the form p/q where p and q are integers. Calculus Eam Thursday November, 998 I Graph the function f) = e 3 /3. Be sure to indicate all asymptotes, all critical points, all points of inflection, all intervals where the function is increasing, decreasing, concave up, concave down. II Consider the function f) = e 3 /3 from the previous problem. Is there a function gy) such that y = f) = gy)? In other words, does the function f have an inverse function?) If yes, give a formula for gy). If no, say why not. III A population of bacteria doubles every two days. How long does it take to triple? The population grows eponentially.) IV whenever The simplest version of l Hospital s rule says that f) a g) = f ) a g ) ) the functions f) and g) are differentiable, ) the derivatives f ) and g ) are continuous, 3) fa) = ga) = 0, and

35 4) g a) 0. Prove this. Give a reason for each step. V Evaluate the following its: sin a) 0 e cos b) 0 c) 0 ) / VI A cylindrical can is made to hold 50 cubic inches of soup. Find the dimensions that will minimize the area of the can. Don t forget that the top and bottom of the can have area as well as the side. The area of the side is the perimeter of the circular top times the height of the can.) VII The hyperbolic functions sinh) = e e, cosh) = e + e satisfy the identity sinh ) cosh ) = constant Specify the constant and prove this identity. b) Find the domain and range of the function f) = cosh). c) Is there a function g such that y = cosh) = gy)? In other words, does the function f) = cosh) have an inverse function?) Why or why not? The domain of g should be the range of f and the range of g should be the domain of f.)

36 I Calculus - Final Eam Evaluate the following: a) dy d where y =. b) dy d where 3 + y + y = 9. n c) n n. Tuesday, December, 998 II a) b) 3 5 c) 3 Evaluate the following: d. d +. d +. III Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the curves y =, y = 4) +, about the line y = 7. Be sure to specify the its of integration in your integral. IV +h h 0 a) Evaluate h d + 3. b) Evaluate the derivative F ) of the function F ) = sin dt + t 3.

37 V The velocity v of a particle at time t is v = e t t. How far does it go from time t = 3 to time t = 5? VI Evaluate 487 k=3 + 7k). Your answer should not contain the summation symbol, but you may leave any other arithmetic undone. You may use the formulas n k = k= nn + ), n k = k= nn + )n + ). 6 VII 5 3 Write a Riemann sum with 00 terms which approimates sin) d. You need not simplify your answer. It should begin 00 with. k= VIII a) A hemispherical bowl of radius 0 inches is filled with water to a height of h inches. How many cubic inches of water are in the bowl? b) If water is being added at a rate of 3 cubic inches per minute, how fast is the water level h rising when the water level is 7 inches? IX Graph y = 4 after answering the following questions: a) On what intervals is the function increasing? b) On what intervals is the function concave up?

38 c) What are the vertical and horizontal asymptotes? X Prove that if w) = u)v) and u) and v) are differentiable, then w ) = u )v) + u)v ). In your proof you may use without proof) the it laws and the fact that a differentiable function is continuous; however, you should indicate where these facts are used in your proof. XI The population of California grows eponentially at a rate of % per year. The population of California on January, 990 was 0,000,000.

39 a) Write a formula for the population Nt) of California t years after January, 990. b) Each Californian consumes pizzas at the rate of 70 pizzas per year. At what rate is California consuming pizzas t years after 990? c) How many pizzas are consumed in California from January, 995 to January, 999?