Podcast Beta
Questions and Answers
What are the even and odd parts of the function $f(x) = x^3 + 5 ext{sin}(x) + 10x^2$?
What is the expression for $I_n$ when $I_n = D^n(x^n ext{log} x)$?
For the limit $lim_{x → 0} (1/x^2 - 1/ ext{sin}^2 x)$, what is the result?
What does the condition $f(2 heta - heta) = f( heta)$ indicate about the curve $r = f( heta)$?
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What does the second derivative test indicate if $f''(x) < 0$?
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Describe how you would find the tangent at the point (1, 2) for the curve defined by the equation 3x² + 2xy + y² - 4 - y = 0 without computing the derivative.
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What relationship can be deduced from the equation Iₙ = Iₙ₋₁ + (n - 1)! regarding the derivatives of the function xⁿ log x?
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Explain how to determine the asymptotes of the curve defined by (y - 1)(x - 2)² + (y + 3x)(y - 2x) + 2x + 2y = 1.
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Discuss how the Leibnitz rule Dⁿ(uv) = Σ(k=0)ⁿ C(n, k) uₖ vₙ₋ₖ applies to the product of two functions.
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What does it mean for a curve y = f(x) to be concave or convex at a point x when considering the second derivative f''(x)?
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Study Notes
Examination Overview
- Mid Semester Examination for B.Sc. Hons. (Mathematics), School of Physical Sciences, Doon University.
- Total time allocated: 1.5 hours.
- Maximum marks: 30.
- Students must attempt four questions from Sections A and B, plus two questions from Section C.
Section A (1.5 x 4 = 6 Marks)
- Odd and Even Parts of Function: Find odd and even components of ( f(x) = x³ + 5 \sin x + 10x² ).
- Compute y₇: Given ( y = x² \sin 5x + e^{2x} ), find the value of ( y₇ ).
- n-th Derivative Calculation: Determine the n-th derivative of ( e^{10x} \sin (4x + 5) ).
- Proof of Integral Iₙ: If ( Iₙ = Dⁿ (xⁿ \log x) ), prove that ( Iₙ = Iₙ₋₁ + (n - 1)! ).
- Power Series Expansion: Expand ( x⁴ - 12 ) in powers of ( (x - 2) ).
Section B (3 x 4 = 12 Marks)
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Proof of Recurrence Relation: From ( y^{1/2} + y^{-1/2} = 2x ), prove that
((x² - 1) y_{n+2} + (2n + 1) x y_{n+1} + (n² - 4) y_n = 0). - Tangent without Derivative: Find the tangent to the curve ( 3x² + 2xy + y² - 4 - y = 0 ) at point (1, 2) without calculating the derivative.
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Asymptotes Finding: Identify all asymptotes for the curve
((y - 1)(x - 2)² + (y + 3x)(y - 2x) + 2x + 2y = 1). -
Limit Calculation: Compute the limit
(\lim_{x \to 0} \left(\frac{1}{x²} - \frac{1}{\sin² x}\right)). - Symmetry Condition Proof: Prove that a curve ( r = f(θ) ) is symmetric about the line ( θ = α ) if ( f(2α - θ) = f(θ) ).
Section C (6 x 2 = 12 Marks)
- Leibnitz Rule Proof: Prove the Leibnitz rule, given by ( Dⁿ(uv) = \sum_{k=0}^{n} C(n, k) u_k v_{n-k} ).
- Concavity and Convexity Proof: Prove a function is concave or convex at point ( x ) based on ( f''(x) < 0 ) or ( f''(x) > 0 ).
- Curve Tracing: Trace the curve defined by ( y²(9 - x²) - x²(9 + x²) = 4 ).
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Description
This quiz is designed for B.Sc. Hons. Mathematics students at Doon University, focusing on key concepts from the MAC-101 Calculus course. Students will tackle multiple-choice questions that cover a range of topics, including function properties, series evaluation, and differential equations. Prepare to test your knowledge and skills in calculus within a 1.5-hour time frame.
