Past Exam Paper 2017-2018 PDF

Preview

Summary

Calculus 1 exam from winter 2017-2018...

Description

UNIVERSITY OF ABERDEEN

SESSION

2017–18

Degree Examination MA1005 Calculus I December 8th 2017

(15:00 - 17:00)

Calculators are not permitted in this examination. Marks may be deducted for answers that do not show clearly how the solution is reached. Attempt ALL FIVE questions from SECTION A and TWO questions from SECTION B. All questions are worth 10 marks. SECTION A – answer all FIVE questions 1.

(2 marks each.) Compute the following limits. x2 + 2x − 8 x→2 x2 + x − 5   2t 1 − (d) lim 2 t→3 t − 9 t−3 (a) lim

2.

(c) lim

x→3

√ x+1−2 x−3

(2 marks each.) Differentiate the functions f , g, h, i and j defined as follows. f (x) =

x +1

2x2

i(x) = ln(x cos(x2 ))

3.

x2 + 2x − 8 x→2 x2 + x − 6 p (sin(θ ))2 + θ 3 (e) lim θ→0 θ

(b) lim

g(t) = t3 ln(t) j(t) = sec(2 ln(t))

h(x) = p 3

1 sin(x) + x2

Let f be the function defined by f (x) = 13 x3 − x2 + 1. (a) (4 marks.) Compute f ′ (x) and f ′′ (x). (b) (3 marks.) Find the critical numbers of f and determine whether they are local maxima or minima. (c) (3 marks.) What are the absolute maximum and minimum values of f on [−1, 1]? (Your answer must state the method being used and explain why it applies.)

4.

(a) (4 marks.) For each of the functions f , g , h and i defined as follows, state whether the function is even, odd, both even and odd, or neither even nor odd. f (x) = x4 + 2x2

g(x) = x5 − x3 + x2 cos(sin(x)) h(x) = cos(x) + (sin(x))2 i(x) = x3 Hint: cos is an even function and sin is an odd function. (b) (3 marks.) Let p be the function defined by p(x) = x2 + x + 1. Use the precise definition of the derivative to show that p′ (x) = 2x + 1. (c) (3 marks.) Let h be the function defined by h(x) = x sin(x/2). Find h(0) and h(π ). Use the intermediate value theorem to show that there is a number c ∈ (0, π) for which h(c) = 3. Your answer should explain why the intermediate value theorem can be applied.

5.

(a) (5 marks.) Use the precise definition of the limit to show that lim (x2 + 2x + 2) = 10. x→2

2

2

(b) (5 marks.) Given that 2x + 3y = 1, use implicit differentiation to show that y′ =

−2x 3y

and

y′′ =

−2 . 9y3

SECTION B – answer TWO questions 6.

(a) (5 marks.) Suppose that f is a differentiable function with domain R = (−∞, ∞), and suppose that f satisfies the condition f ′ (x) = f (x2 ) for all x. Show that f ′′ (x) = 2x · f (x4 ), and that f ′′′(x) = 2f (x4 ) + 8x4 f (x8 ), and that f (4) (x) = 40x3 f (x8 ) + 64x11 f (x16 ) for all x. (b) (5 marks.) Let p be the function defined as follows.  x · |x| · cos( 1x ) if x 6= 0 p(x) = 0 if x = 0 Use the precise definition of the derivative to show that p′ (0) = 0. Hint: In answering this question you may need to use the squeeze theorem.

7.

(a) (5 marks.) Use the precise definition of the limit to show that lim

x→5

x+3 = 2. x−1

(b) (5 marks.) Give an example of one function f which satisfies all three of the following properties. • f (x) = 0 for x 6 0 • f (x) = 1 for x > 1 • f is differentiable with domain (−∞, ∞) You must show that the function satisfies the three conditions.

8.

(a) (5 marks.) Let f be the function defined as follows.  x+1 if 1 6 x f (x) = bx2 + c if x < 1 For which values of b and c is f differentiable at 1? Your answer should give the values of b and c and should show clearly that f is differentiable at 1 for these choices of b and c. (b) (5 marks.) Let f be the function defined by f (x) = bx − 21ax2 , where a and b are constants satisfying 0 < b < a. Find the maximum and minimum values of f on the interval [−1, 1]. Your answers for the maximum and minimum values should be formulas involving a and b but not x.

MA1005 Calculus I

December 8th 2017...