Math 131 Final Exam Spring 2016

Transcription

1 Math 3 Final Exam Spring 06 Name: ID: multiple choice questions worth 5 points each. Exam is only out of 00 (so there is the possibility of getting more than 00%) Exam covers sections. through 5.4 No graphing calculators! Any non-graphing, non-differentiating, non-integrating scientific calculator is fine. Mark your answers on the answer card. sin(a ± B) = sin A cos B ± sin B cos A tan(a ± B) = tan A ± tan B tan A tan B sin (A/) = cos A sin A sin B = [cos(a B) cos(a + B)] cos cos(a ± B) = cos A cos B sin A sin B tan(a/) = cos A sin A = sin A + cos A cos (A/) = + cos A A cos B = [cos(a B) + cos(a + B)] sin A cos B = [sin(a + B) + cos(a B)] ( ) ( ) A + B A B sin A + sin B = sin cos ( ) ( ) A + B A B cos A + cos B = cos cos Law of Cos: c = a + b ab cos C n(n + ) i = ( n(n + ) i 3 = ( A + B sin A sin B = cos ( A + B cos A cos B = sin Law of Sin: sin A a i = = sin B b ) sin ) sin n(n + )(n + ) 6 ) V Sphere = 4 πr3 ( ) A B ( ) A B = sin C c V Cone = 3 πr h V Pyramid = 3 Bh

2 Math 3 Final Exam Page of??. Suppose f(4) =, g(4) = 3, f (4) = 6, and g (4) =. Let h(x) = Find h (4). g(x) f(x) + g(x) A. Derivative does not exist or is unable to be determined B. 6 C. D. 8 E. 4 F. 0 G. 4 H. 8 I. J. 6 Solution: Quotient rule h (x) = g (x)(f(x) + g(x)) g(x)(f (x) + g (x)) (f(x) + g(x)) h (4) = g (4)(f(4) + g(4)) g(4)(f (4) + g (4)) (f(4) + g(4)) ( )( 3) ( 3)(6 + ( )) = = 6 ( 3). Suppose f and g are continuous on [a, b], a < c < b, and f(x) g(x). Which of the following statements are always true? I. II. b a b (f(x) g(x)) dx = f(x) dx b a a b g(x) dx f(x) dx b a a g(x) dx

3 Math 3 Final Exam Page 3 of?? III. c a f(x) dx A. None b B. I only C. II only D. III only E. I and II only F. I and III only f(x) dx = c a b f(x) dx G. II and III only H. I, II and III Solution: These are all properties of integrals from Section 5. of the text. III is a slight rearrangement of the property b a f(x) dx + b c f(x) dx = b a f(x) dx

4 Math 3 Final Exam Page 4 of?? 3. If f and g are differentiable, which of the following statements are always true? I. d dx (f(x) + g(x)) = f (x) + g (x) II. d ( ) f(x)g(x) = f (x)g (x) dx III. d dx f( g(x) ) = f ( g(x) ) g (x) IV. d f (x) f(x) = dx f(x) A. None B. Exactly one of these is true C. Exactly two of these is true D. I, II and III only E. I, II and IV only F. I, III and IV only G. II, III and IV only H. All are true Solution: II is not true. The true statement is the product rule. 4. Let f(x) = x 3 6x + 5. Let m be the absolute minimum of f(x) on [ 3, 5]. absolute minimum occurs. Find the x-value of where this A. There is no absolute minimum on this interval. B. 3 C. D. E. 0 F. G. H. 3

5 Math 3 Final Exam Page 5 of?? I. 4 J. 5 K. 00 Solution: f (x) = 3x x = 3x(x 4). The critical points are x = 0 and x = 4. Plugging in the points: x y 3 76 MIN 0 5 MAX

6 Math 3 Final Exam Page 6 of?? 5. A function, f(x), has derivatives: f (x) = (x + ) 3 (x + ) f (x) = (5x + 8)(x + ) (x + ) Let: M = Number of local maximum m = Number of local minimum I = Number of inflection points Find M + m + I A. 0 B. C. D. 3 E. 4 F. 5 G. 6 Solution: Make charts for f and f. f f + x < < x < 8/5 8/5 < x < x > Thus, the only local extrema is at x =, which is a minimum. x =, 8/5 are both inflection points. M = 0, m =, I =. M + m + I = Let f(x) = x + x. Find f (). A. B.

7 Math 3 Final Exam Page 7 of?? C. D. E. F. G Solution: f (x) = (x + ( x) / + ) x / ( ) ( = + ) x + x x / f () = ( ) ( + ) + () / = 3 4

8 Math 3 Final Exam Page 8 of?? 7. If g(x) = x + f(x), where f() = and f () = 3. Find g (). A. g () does not exist B. 7 C. 0 D. 7/8 E. 7/4 F. 7/ G. 7 Solution: g (x) = (x + f(x)) / ( + f (x)) g () = ( + ()) / ( + (3)) = (4) / (7) = Find the derivative of f(x) = x e /x. A. f(x) is not differentiable B. x e /x C. xe /x D. e /x E. F. x e /x x e /x G. e /x (x ) H. e /x (x + ) Solution: f (x) =x(e /x ) + x e /x (/x ) =e /x (x + )

9 Math 3 Final Exam Page 9 of?? 9. Find the derivative of f(x) = x x. A. x x B. x x (ln x) C. x x (ln x + ) D. x x (ln x ) E. x x ln x x x F. ln x + G. x x ln x Solution: f(x) =exp (ln x x ) = exp (x ln x) ( f (x) =exp (x ln x) ln x + x ) x f (x) =x x (ln x + )

10 Math 3 Final Exam Page 0 of?? 0. If f(x) = ln(x + ln x). Find f (). A. 0 B. / C. ln D. E. ln 3 F. 3/ G. H. 5/ I. 3 Solution: ( f (x) = + ) x + ln x x f () = ( + ) = 3 + ln. Find the slope of the tangent line to the curve: sin(x + y) = x y at the point (π, π). A. 3 B. 3 C. 3 D. 3 E. F. G. H. I.

11 Math 3 Final Exam Page of?? J. Solution: Take the derivative implicitly. cos(x + y)( + y ) = y y = cos(x + y) + cos(x + y) y (π, π) = cos(π) + cos(π) = + = 3

12 Math 3 Final Exam Page of??. Evaluate the integral: (5x 4 + 3x )dx A. 8 B. 54 C. 7 D. 00 E. 34 F. 5 G. 000 H. 0, 000 Solution: (5x 4 + 3x )dx = 3x 5 + x 3 = 3(3) + 8 (3() + ) = Find lim x 0 e x e x x x sin x A. Does not exisit B. C. 0 D. E. F. 3 G. 4 H.

13 Math 3 Final Exam Page 3 of?? Solution: e x e x x lim x 0 x sin x LH = lim x 0 e x + e x cos x LH = lim x 0 e x e x sin x LH = lim x 0 e x + e x cos x =

14 Math 3 Final Exam Page 4 of?? 4. At x =, a function f(x) has tangent line y = 4x + 3. Find f ( ) + f( ). A. It is impossible to determine B. C. 0 D. E. F. 3 G. 4 H. 5 I. 00 Solution: Let s name the tangent line: L(x) = 4x + 3. At x =, we must have f( ) = L( ) and f ( ) = L ( ). L( ) = 4( ) + 3 = and L ( ) = 4. Thus, f ( ) + f( ) = L ( ) + L( ) = 4 + ( ) = Which of the following statements are always true? I. II. III. IV. = n (a i b i ) = a i b i ( )( ) (a i b i ) = a i b i i = n(n + ) A. None are always true B. Exactly one is always true C. Exactly two are always true D. Only I, II and III

15 Math 3 Final Exam Page 5 of?? E. Only I, II and IV F. Only I, III and IV G. Only II, III and IV H. All are always true Solution: I, II and IV were discussed in class. You can test that III is false by testing some examples. Here s one, let a i = and b i =, then ()() =n ( ) ( ) =n

16 Math 3 Final Exam Page 6 of?? 6. Estimate the area under the graph of f(x) = + x from x = to x = using three rectangles and right endpoints (a right hand sum). A. 3 B. 4 C. 5 D. 6 E. 7 F. 8 Solution: x = ( ( ))/3 =. x 0 =, x =, x = 0, x 3 =. RHS =f(x ) x + f(x ) x + f(x 3 ) x =f( )() + f(0)() + f()() =( + ) + ( + 0) + ( + ) = 5 7. Estimate the area under the graph of f(x) = + x from x = to x = using three rectangles and left endpoints (a left hand sum). A. 3 B. 4 C. 5 D. 6 E. 7

17 Math 3 Final Exam Page 7 of?? F. 8 Solution: x = ( ( ))/3 =. x 0 =, x =, x = 0, x 3 =. LHS =f(x 0 ) x + f(x ) x + f(x ) x =f( )() + f( )() + f(0)() =( + 4) + ( + ) + ( + 0) = 8

18 Math 3 Final Exam Page 8 of?? 8. Evaluate the integral x 3 dx A. 3 B. 3 C. 3 D. 3 E. 0 F. 3 G. 3 H. 3 I. 3 Solution: x 3 dx = x3 3 = 0 9. Express the limit as a definite integral. lim 3 ( n cos 4 + 3i ) n A. B. C. D cos x dx cos( 4 + x) dx cos( 4 + x) dx cos x dx

19 Math 3 Final Exam Page 9 of?? E. F. G. H sin x dx sin( 4 + x) dx sin( 4 + x) dx sin x dx Solution: You can see that x = 3 n. x i = a + i x. Matching up with the given sum, it looks like a = 4. This means f(x) = cos x and b =.

20 Math 3 Final Exam Page 0 of?? 0. Find the values of a and b that make f continuous everywhere. x 4 if x < x f(x) = ax bx + 3 if x < 3 x a + b if x 3 What is a + b? A. It is impossible to find any such a and b. B. 0 C. / D. E. F. 3 Solution: lim f(x) =4 x lim f(x) =4a b + 3 x + lim f(x) =9a 3b + 3 x 3 lim f(x) =6 a + b x 3 + This gives us the equations 4 = 4a b + 3 and 9a 3b + 3 = 6 a + b Solving for a and b gives a = / and b = /.. Express the integral as a limit of sums π/3 0 A. lim sin(3x) dx ( ) π iπ n sin n

21 Math 3 Final Exam Page of?? B. lim C. lim D. lim E. lim F. lim G. lim H. lim ( ) π iπ 3n sin 3n π 3n sin ( ) π 3iπ n sin n ( ) 3π 3iπ n sin n ( ) 3π iπ n sin n ( ) π 3iπ n sin n ( ) π 3iπ 3n sin n ( ) iπ n Solution: x = π 3n. x i = iπ 3n. Thus, π/3 0 sin 3x dx = lim f(x i ) x = lim ( ) π 3iπ 3n sin = lim 3n ( ) π iπ 3n sin n