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Calculus Exercises: Algebraic Expressions, Limits, Differentiation, and Integration, Exams of Mathematics

Aliah UniversityMathematics

A set of calculus exercises covering various topics such as algebraic expressions, limits, differentiation, and integration. It includes simplification of algebraic expressions, solving quadratic equations, graphing functions, finding inverse functions, sum and sum of infinite geometric series, evaluating limits, differentiating various functions, finding the equation of tangent lines, and evaluating definite and indefinite integrals.

Typology: Exams

2012/2013

Uploaded on 02/26/2013

dharamnishth
dharamnishth ๐Ÿ‡ฎ๐Ÿ‡ณ

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SECTION A
1. Simplify:
(a) a9bโˆ’3c8
a7(bc2)4(b) 16x2โˆ’1
16x2+ 8x+ 1.
[4 marks]
2. Write 1
yโˆ’7โˆ’7
y2โˆ’7yas a single fraction, and simplify it as far as possible.
[4 marks]
3. Solve the following quadratic equations:
(a) x2โˆ’xโˆ’72 = 0 (b) 8x2โˆ’2xโˆ’21 = 0.
[4 marks]
4. Sketch the graph of each of the functions:
(a) y= 3xโˆ’6 (b) y=x2+ 6x+ 8 (c) y=|x2+ 6x+ 8|.
[7 marks]
5. Given that f(x) = 2x+ 7
1โˆ’3x, obtain an expression for the inverse function
fโˆ’1(x).
[3 marks]
6. (a) Find the sum of the geometric series
5
X
n=1
(โˆ’5)n.
(b) Write down the formula for the sum of the infinite geometric series
โˆž
X
n=1
arnโˆ’1with first term aand common ratio r, when |r|<1.
Hence show that โˆž
X
n=1 ๎˜7
10 ๎˜‘n=7
3.
[6 marks]
Paper Code MATH 011 Sept-06 Page 2 of 5 CONTINUED
pf3
pf4

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Download Calculus Exercises: Algebraic Expressions, Limits, Differentiation, and Integration and more Exams Mathematics in PDF only on Docsity!

SECTION A

  1. Simplify: (a) (^) aa (^79) (bbcโˆ’^32 c)^84 (b) (^16) x^162 x+ 8^2 โˆ’x^1 + 1. [4 marks]
  2. Write (^) y โˆ’^1 7 โˆ’ (^) y (^2) โˆ’^7 7 y as a single fraction, and simplify it as far as possible. [4 marks]
  3. Solve the following quadratic equations: (a) x^2 โˆ’ x โˆ’ 72 = 0 (b) 8 x^2 โˆ’ 2 x โˆ’ 21 = 0. [4 marks]
  4. Sketch the graph of each of the functions: (a) y = 3x โˆ’ 6 (b) y = x^2 + 6x + 8 (c) y = |x^2 + 6x + 8|. [7 marks]
  5. Given that f (x) =^21 x โˆ’^ + 7 3 x, obtain an expression for the inverse function f โˆ’^1 (x). [3 marks]
  6. (^) (a) Find the sum of the geometric seriesโˆ‘^5 n=1^ (โˆ’5)

n (^). โˆ‘^ โˆž (b)^ Write down the formula for the sum of the infinite geometric series n=1^ ar

nโˆ’ (^1) with first term a and common ratio r, when |r| < 1. Hence show thatโˆ‘ n^ โˆž=1^ (^107 )n = 73.

  1. Evaluate the following limits: (a) (^) nlimโ†’โˆž 5 n^22 n (^) + 6^2 โˆ’n^7 nโˆ’ 1 (b) lim xโ†’ 4 x^2 xโˆ’ (^2) โˆ’x^ โˆ’ 16 12. [4 marks]
  2. Differentiate with respect to x (a) (3x โˆ’ 8)^7 (b) (x^4 + 3)^54 (c) x^8 cos x. [8 marks]

the point where^ 9.^ Write down the equation of the tangent line to the curve x = โˆ’1.^ y^ = 3x^4 + 2 at [3 marks]

  1. Find the indefinite integrals: (a)^ โˆซ (6 โˆ’ 3 x^5 โˆ’ sin x) dx (b)^ โˆซ eโˆ’^5 x^ dx. [6 marks]
  2. Evaluate the definite integrals: (a)^ โˆซ^0 ฯ€/^14 cos 7x dx (b)^ โˆซ^365 x โˆ’^5 14 dx [Substitute u = 5x โˆ’ 14].
  1. Find the indefinite integrals: (a)^ โˆซ x^5 cos(x^6 + 4) dx [Substitute u = x^6 + 4] (b)^ โˆซ tan^8 x sec^2 x dx [Substitute t = tan x]. Evaluate the definite integral: (c)^ โˆซ^

โˆš 2 / 5 0 โˆš^ dx 4 โˆ’ 25 x^2 [Substitute^ x^ =

5 sin^ t].