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Calculus Exam for MATH 011, Exams of Mathematics

Avinashilingam UniversityMathematics

This is a calculus exam for math 011, which covers simplification of expressions, solving quadratic equations, graphing functions, finding inverse functions, summing arithmetic and geometric progressions, evaluating limits, differentiating and integrating functions, and finding tangent lines and indefinite integrals. It also includes a relation between x and y, and its derivatives, and finding the area bounded by a curve and the x-axis.

Typology: Exams

2012/2013

Uploaded on 02/26/2013

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MATH 011 Jan-11: Calculus
Examiner: Dr. Th. Eckl, Extension 44051.
Time allowed: Three hours
All answers to Section A and the best THREE answers to Section B will be
counted.
The marks shown against questions, or parts of questions, indicate their rela-
tive weight.
Section A carries 55% of the available marks.
Paper Code MATH 011 Jan-11 Page 1 of 7 CONTINUED
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MATH 011 Jan-11: Calculus

Examiner: Dr. Th. Eckl, Extension 44051.

Time allowed: Three hours All answers to Section A and the best THREE answers to Section B will becounted.

The marks shown against questions, or parts of questions, indicate their rela-tive weight.

Section A carries 55% of the available marks.

SECTION A

  1. Simplify: (i) (^) a(โˆ’ab (^1) ()bc^3 c (^25) ) 3 (ii) (^) 3 + 4^1 โˆ’x^ x+^2 x 2. [4 marks]
  2. Write (^) b (^2 2) + 2^ โˆ’^ b^2 โˆ’b 3 โˆ’ (^) b (^2) + 3^6 b as a single fraction, and simplify it as far as possible. [4 marks]
  3. Solve the following quadratic equations: (i) 11 x โˆ’ 40 + 2x^2 = 0. (ii) 14 x^2 + 5x โˆ’ 39 = 0. [4 marks]
  4. Sketch the graph of each of the functions: (i) y = โˆ’ 2 x + 4 (ii) y = x^2 โˆ’ 5 x + 4 (iii) y = |x^2 โˆ’ 5 x + 4|. [7 marks]
  5. Given that f (x) =^45 xโˆ’ + 2^3 x, obtain an expression for the inverse function f โˆ’^1 (x). [3 marks]

initial term -100 and the common difference 2.^ 6.^ Find the sum of the first 80 terms of the arithmetic progression with the [2 marks]

  1. Find the sums of the geometric series: (i) nโˆ‘=0^7 ( โˆ’^12 )n (ii) โˆ‘ n^ โˆž=1^ (^37 )n. [4 marks]

SECTION B

  1. Suppose that x and y satisfy the following relation: 2 x^3 + 3xy^2 + 6y^4 = 1. dy^ (i)^ By differentiating both sides of this equation with respect to^ x^ express dx in terms of(ii) Use the result of (i) to find the tangent line to the curve^ x^ and^ y. 2 x^3 + 3xy^2 + 6y^4 = 1 at the point (โˆ’ 1 , 1). (iii) Show that this line is also tangent to the parabola y = โˆ’^34 x^2 + x โˆ’ (^14) in the point (1, 0). [15 marks]
  2. (a) Differentiate the following functions with respect to x: (i) x 2 2 โˆ’e^3 x 3 โˆ’x^23 (ii) sin(4x โˆ’ 3) cos^2 (2x + 1) [9 marks] (b) Find the indefinite integrals: (i)^ โˆซ x^5 ex^6 โˆ’^3 dx (ii)^ โˆซ^ tan cos 23 33 xx dx. [6 marks]
  3. Find the first and second derivatives of the function f (x) = (^5) โˆ’^2 3 x + (^2) x + 10^3. Find all the stationary points and determine their nature.and vertical asymptotes of the graph of y = f (x). Hence sketch the graph. Find all horizontal

[15 marks]

  1. (a) Evaluate the definite integrals: (i)^ โˆซ^ โˆ’ฯ€/ฯ€/^86 tan 2x dx (ii)^ โˆซ^053 25 + 9^ dxx 2 [ Substitute x by^53 tan t. ] [6 marks] x-axis. Calculate the total area bounded by the curve and the(b)^ Find the three points at which the curve^ y^ =^ x^3 + 3x 2 x^ โˆ’-axis.^4 x^ crosses the [9 marks]
  1. Trigonometric functions tan x = (^) cossin^ xx cot x = cos sin^ xx sec x = (^) cos^1 x cosec x = (^) sin^1 x
  2. Derivatives (a) If y = xn^ where n is constant then dy dx = nxnโˆ’^1. (b) If y = uv, where u and v are functions of x, then dy dx = u (^) dxdv + v du dx. (c) If y = u v , where u and v are functions of x, then dy dx = v^ du^ dx^ vโˆ’ 2 u^ dv^ dx. (d) If y is a function of u and u is a function of x then dy dx = dy du ร— du dx. (e) (^) dxd (yr) = (^) dyd (yr) ร— dy dx = ryrโˆ’^1 dy dx. Function Derivative sin x cos x cos x โˆ’ sin x tan x sec^2 x ln x (^1) x ex^ ex
  3. Integrals โˆซ โˆซ xn^ dx^ =^ n^1 +1^ xn+ (^1) x dx = ln x โˆซ ex (^) dx = ex โˆซ cos x dx = sin x โˆซ sin x dx = โˆ’ cos x โˆซ sec (^2) x dx = tan x

Paper Code MATH 011 Jan-11 Page 7 of 7 END