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October 2024 Question Paper

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MAT1CJ101 — Differential Calculus

End Semester Examination — October 2024

Programme: B.Sc. Mathematics Honours (CUFYUGP 2024) University: University of Calicut Semester: I | Credits: 4 Time: Two Hours | Maximum Marks: 70

Exam Pattern

Section Type Marks each Ceiling Strategy
A Short questions 3 24 Answer as many as you can
B Medium questions 6 36 Answer as many as you can
C Long questions 10 10 Answer any one of two

Total: 70 marks (24 + 36 + 10)

Section A

Answer any number of questions. Each carries 3 marks. Overall ceiling: 24 marks.

  1. Determine the domain of the composite function f∘g of the functions f(x) = √x and g(x) = x + 1. Evaluate f at the points g(3) and f(9).

  2. Evaluate lim(x→0) (√(x+2) − √2) / x.

  3. Does the curve y = x⁴ − 2x² + 2 have any horizontal tangents? If so, where?

  4. The curve y = ax² + bx + c passes through the point (1, 2) and is tangent to the line y = x at the origin. Find a, b, and c.

  5. Find dy/dx if 2y = x² + sin y.

  6. Find the normal to the curve x² − xy + y² = 7 at the point (−1, 2).

  7. Find the absolute extrema of f(x) = x² − 3 on [−2, 3).

  8. If f′(x) = 0 at each point of an interval I, then show that f(x) = C for all x in I, where C is a constant.

  9. Give an example of a function defined on [0, 1] that has neither a local maximum nor a local minimum value at 0.

  10. Show that lim(x→∞) 1/x = 0.

Section B

Answer any number of questions. Each carries 6 marks. Overall ceiling: 36 marks.

  1. Give an equation for the shifted graph of x = 3y² up 2 and right 3 units. Then sketch the original and shifted graphs together.

  2. Is any real number exactly 1 less than its cube? Justify your answer.

  3. Define the left-hand limit of a function f at a point x₀. Give one example.

  4. Find the average rate of change of f(t) = 1/t with respect to t over the interval from t = 2 to t = 3.

  5. What is implicit differentiation? When do you need it? Give examples.

  6. Show that the function f(x) = x⁴ + 3x + 1 has exactly one zero in the interval [−2, −1].

  7. Using the Sandwich Theorem to find the asymptotes of the curve y = 2 + (sin x)/x.

  8. Find a function that satisfies the following conditions and sketch its graph:

    • lim(x→∞) f(x) = 1
    • lim(x→−∞) f(x) = −1
    • lim(x→1⁺) f(x) = 1

Section C

Answer any one question. The question carries 10 marks.

  1. (a) Find the intervals on which f(x) = x³ + 12x + 5, −3 ≤ x ≤ 3 is increasing and decreasing. Where does the function assume extreme values and what are these values?

    (b) Show that f(x) = (x² + x − 6)/(x² − 4) has a continuous extension to x = 2, and find that extension.

  2. Graph the function y = (x³ + 1)/x.

Module-wise Question Map

Q Module Topic Section Style
1 I Domain of composite functions A (3 marks)
2 I Limit by rationalizing A (3 marks)
11 I Shifting graphs B (6 marks)
12 I/III Intermediate Value Theorem B (6 marks)
13 I Left-hand limit — definition B (6 marks)
3 II Horizontal tangents (derivative = 0) A (3 marks)
4 II Tangency condition — system of equations A (3 marks)
5 II Implicit differentiation A (3 marks)
6 II Implicit diff — normal to curve A (3 marks)
14 II Average rate of change B (6 marks)
15 II Implicit differentiation explained B (6 marks)
7 III Absolute extrema on interval A (3 marks)
8 III Consequence of zero derivative (MVT) A (3 marks)
9 III Example with no local extremum at endpoint A (3 marks)
16 III IVT + uniqueness via MVT B (6 marks)
19a III Increasing/decreasing + extrema C (10 marks)
19b III/II Continuous extension C (10 marks)
10 IV Limit at infinity A (3 marks)
17 IV Asymptote via Sandwich Theorem B (6 marks)
18 IV Constructing function from asymptotic behavior B (6 marks)
20 IV Full curve sketching C (10 marks)
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