MATH 110 University at Buffalo Wk4 Calculus Questions

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Math 110 - Calculus I Section I- Multiple Choice Students Name Questions 1 and 2 refer to the diagram below. y 4 3 2 1 x –3 –2 –1 1 3 2 –1 4 5 6 7 –2 –3 1. What is the value of lim f (x) . x5 B. – 1 A. Does not exist 2. What is the value of C. 4 D. 1 E. Some other answer C. 2 D. 1 E. Some other answer lim f (x) . x1 A. Does not exist B. – 1 x2 9 3. What is the value of the lim x3 A. 6 B. 0 x 3 . C.  D. – 3 E. Some other answer Page 1 x 4. 5. For what value(s) of x is the function f(x) = 6. C. x = 1 and x = – 1 B. x = 1 D. x = 0 , x = 1 and x = – 1 E. Some other answer 3 What is the value of the following limit lim 4x 1 x2x 2 2 B. 0 C.  D.  B. ½ C.  D.  1 B. 0 C.  D.  2 8. E. Some other answer 3 lim 2x 1 x4x3 2 What is the value of the following limit A. E. Some other answer lim 4x 1 x2x 2 What is the value of the following limit A. 2 7. x 1 A. x = 0 A. 2  not continuous? 2 The equation of the tangent line to the curve y = 2 x  2 E. Some other answer at the point (1,0) is:- x 9. A. y = 2x – 1 B. y = 2x + 1 D. y = 2 E. Some other answer C. y = 2x The equation of the tangent line to the curve y = x3 – 5x + 3 at the point (2,1) is:A. y = 8x + 13 B. y = – 9x – 13 D. y = –7x + 13 E. Some other answer 10. What is the derivative of the function y = x   A. 1  2 x D. 1 2 x 1 2 x3  1  1 2 2 B. 1 2 x 1 x C. y = 7x – 13  2  1 2 x3 1  2 2 C. 1 2 x  1 2 x3 E. Some other answer 2 x3 Page 2 11. The position of a particle is given by the formula s(t) = 1 cos (2 t) (with t 0 ). 4 What is the acceleration when t = 5 A. B. 2 0 D. – 2 C. – 4 2 E. Some other answer 12. What is the derivative of y = x2 e2x A. 2xe2x + x2ex B. 2xe2x + 2x2e2x D. xe2x + 2x2e2x E. Some other answer C. xe2x + x2e2x t 2 1 13. What is the derivative of g(t) = t 1 t 2 t 1 A. 1 D. B. 2t t 2 t 1 C. E. Some other answer t 12  14. What is the derivative of f(x) = sin( 2x 1) A. cos  2x 1  B. 2x 1  2x 1 D. cos t 12  2 cos 2x 1  2x 1  C. cos  2x 1  2 2x 1  E. Some other answer 15. For h(x) = x3 + 5x2 – e4x what is h(x) the third derivative of h(x)? A. 6 – e4x B. 6 + e4x D. 6 – 64e4x E. Some other answer C. 6 Page 3 16. The position of a particle is given by the formula s(t) = t3 – 1.5t2 – 2t (with t 0 ). What is the acceleration when t = 5 A. 27 m/sec2 C. – 27 m/sec2 B. 35 m/sec2 D. – 35 m/sec2 E. Some other answer 17. The position of a particle is given by the formula s(t) = 1 cos (2 t) (with t 0 ). 4 What is the acceleration when t = 0 A. B. 2 0 D. – 2 C. – 4 2 E. Some other answer 18. Use logarithmic differentiation to differentiate the function y = x6x. A. dy = x6x B. dx D. dy dy = 6x6x(6ln x + 1) dx = x6x(6ln x + 1) C. dy = 6x6x(ln x + 1) dx E. Some other answer dx 19. For the function y = x what is the equation of the linear approximation function at x = 4 A. y = 2 B. y = ¼ x C. y = ¼ x + 1 D. y = ¼ x + 2 E. Some other answer 20. Find the absolute minimum value of y = x3 – 3x on the interval [0,2]. A. 0 C. – 2 B. 2 D. – 3 E. Some other answer 21. Find all the critical numbers for f(x) = x4(x – 3)3 A. x = 0 , x = 2 and x = C. x = 0 , x = 3 and x = 12 7 12 7 B. x = 0 , x = 3 and x = D. x = 0 , x = 2 and x = 12 11 12 11 E. Some other answer Page 4 22. On what interval is f(x) = x4 – 6x2 concave down. A. (– 1, 1) B.(0,1) C. (– 1, 0) D. ( , 1) E. Some other answer 23. If f (1) = 0 and f (1) – 2 then which of the following is a logical conclusion A There is a local maximum at x = 1 B There is a local minimum at x = 1 C There is a point of inflection at x = 1 D There is a vertical asymptote at x = 1. E. Some other answer. 24. If f (x)  f(x) which one of the following statements is true. B. f(x) = ex + cx A. f(x) = x D. It is impossible for a function to have C. f(x) = ex + cx + d f (x) f(x) E. None of the statements are true. 25. If f (x) 3x2 4x 1 and f(0) = 1 Then A. f(x) = x3 + 2x2 + 1 B. f(x) = x3 + 2x2 + x + 1 D. f(x) = 6x + 1 E. Some other answer. C. f(x) = 6x + 4 26. If f (1) = 0 and f (1) – 2 then which of the following is a logical conclusion A There is a local maximum at x = 1 B There is a local minimum at x = 1 C There is a point of inflection at x = 1 D There is a vertical asymptote at x = 1. E. Some other answer. 27. Find the absolute maximum value of y = A. 10 B. 81 C. 8 81 x 2 on the interval [– 9,9]. D. 1 E. Some other answer 28. Find f(x) when f (x) = 12x + 24x2 A. f(x) = 6x3 + 4x4 + cx + d B. f(x) = 2x3 + 2x4 + cx + d C. f(x) = 4x3 + 8x4 + cx + d D. f(x) = 2x3 + x4 + cx + d E. Some other answer Page 5 Section II. In this section it is crucial to show relevant working as credit can only be given for working that is shown. You must also answer using the method asked in those questions that specify a particular method. 1. Find the following limits by algebraic means – Do not use L’Hospitals Rule. (a) 2 lim x 4 x2 x 2 1 lim 1h (b) h0 h 2. By using limits and first principals find f (a) for the function f(x) = x2 – x. Page 6 3. Draw a sketch of y = f (x) by using the graph of y = f(x). y = f(x) 4. Differentiate the functions:(a) f(x) = 2x3 + 4 + 2 x 4.(b) h(x) = 4 e x x2 Page 7 5. 4.(c) g(x) = 1 x2 4.(d) f(x) = sin3(2x + 1) 4.(e) h(x) = x3 4.(f) e x x g(x) = x e x Find dy x 2 1 for the function x2y2 + 4y = 5 by using implicit differentiation. dx Page 8 6. At noon ship A is 200 km west of ship B. Ship A is sailing north at 40 km/hour while ship B is sailing north at 25 km/hour. A 2 Give a formula for D in terms of x and y. (a) D x B y 200 km dx and dy , x , y and D when the time is 6:00 p.m. (b) What are the values of (c) What is the rate of change of the difference D between the ships at 6:00 p.m. dt dt 7. The volume of a spherical balloon is V = 4 r 3 . 3 Page 9 When the radius is 10 cm the radius is increasing at the rate of 2 cm/sec ( What is the rate of increase in the volume ( 8. dV dr 2) dt ) when the radius is 10 cm. dt A farmer has 1000 feet of fencing; he wishes to enclose a rectangular area, partitioned into 4 as shown below. y x (a) Use the above information to find an expression for y, in terms of x. (b) Find a mathematical formula for the area of the rectangle in terms of x. (c) What value of x gives the maximum area for the above rectangle 9. Find the limits of the following using L’Hospitals Rule. Page 10 (a) (b) 3 lim x 2 27 x3 x 9 4 lim x 2256 x 16 x4 (c) lim sin x cos x 1 x0 sin x cos x 1 (d) lim x ln(x) x0  10. The function f(x) = x3 – 3x + 7. Page 11 (a) For what intervals is the function f(x) increasing and decreasing? (b) Where are the local max and min , if they exist? (c) When is f(x) concave up, concave down and where are the points of inflection, if they exist? 11. A farmer has 100 feet of fencing, he can use a building on one side but he must fence the other three sides. What are the dimensions of the rectangle PQRS that has the greatest area. S P y x Q R Page 12 12. The derivative f (x) cos x 4x 2 , if you are also told that f(0) = 2. Find a function f(x) that has all these properties. 13. Use the following information f (1) 2 , f (3) 6 g'(1) 1 , g'(2) 3 m(x) = f(x)g(x) , f (1) 4 and f (3) 10 g(1) 3 and g(3) 1 f (x) r(x) = and p(x) = f(g(x)) g(x) To find m'(1), r'(1) and p'(1) Page 13 Page 14
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Explanation & Answer

Hi - this is done!

A. The limit does not exist.

D. 1

A. 6

C. x=1, x=-1

C. −∞

A. 2

E. Some other answer

E. Some other answer (y=2x-2)

C. Y=7x-13

𝑫.

D. −𝝅𝟐

𝟏
𝟐√ 𝒙

+

𝟏
𝟐√𝒙𝟑

B. 𝟐𝒙𝒆𝟐𝒙 + 𝟐𝒙𝟐 𝒆𝟐𝒙

E. Some other answer

A.

𝑪𝒐𝒔(√𝟐𝒙+𝟏)
√𝟐𝒙+𝟏

D. 𝟔 − 𝟔𝟒𝒆𝟒𝒙

A. 27 m/sec^2

D. −𝝅𝟐

B.

𝒅𝒚
𝒅𝒙

= 𝟔𝒙𝟔𝒙 (𝒍𝒏𝒙 + 𝟏)

C. Y=1/4x +1

C. -2

E. Some other answer

A. (-1,1)

B. There is a local minimum at x=1

E. None of the statements are true

B. There is a local minimum at x=1

E. Some other answer

B. 𝟐𝒙𝟑 + 𝟐𝒙𝟒 + 𝒄𝒙 + 𝒅

Section 2
1. Find the limits by algebraic means
a.

𝐥𝐢𝐦

(𝒙𝟐 −𝟒)

𝒙→ −𝟐 (𝒙+𝟐)

b. 𝐥𝐢𝐦
𝒉→𝟎

√𝟏+𝒉−𝟏
𝒉

= 𝐥𝐢𝐦

(𝒙−𝟐)(𝒙+𝟐)
(𝒙+𝟐)

𝒙→ −𝟐

= 𝐥𝐢𝐦

√𝟏+𝒉−𝟏

𝒉→𝟎

𝒉



= 𝐥𝐢𝐦 (𝒙 − 𝟐) = −𝟒
𝒙→ −𝟐

√𝟏+𝒉+𝟏
√𝟏+𝒉+𝟏

𝟏+𝒉−𝟏

= 𝐥𝐢𝐦 𝒉(√𝟏+𝒉+𝟏) = 𝐥𝐢𝐦
𝒉→𝟎

𝟏

𝒉→𝟎 √𝟏+𝒉+𝟏

𝟏

=𝟐

2. Find f�...


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