Continuity and Differentiability — Questions & Answers

Sample questions from Continuity and Differentiability

1. Q1. If y_n denotes the nth derivative of y = e^(2x)*sin(3x), then y_1 at x = 0 equals:

   • A.3✓
   • B.2
   • C.5
   • D.6
   Solution

   y' = e^(2x)(2 sin3x + 3 cos3x); at x=0 this is 3.

2. Q2. If y = x^(cos x), then dy/dx equals:

   • A.x^(cos x)*(cos x/x + sin x * ln x)
   • B.x^(cos x)*(-sin x)
   • C.cos x * x^(cos x - 1)
   • D.x^(cos x)*(cos x/x - sin x * ln x)✓
   Solution

   ln y = cos x ln x; y'/y = -sin x ln x + cos x/x.

3. Q3. The value of k that makes f(x) = (sqrt(4 + x) - 2)/x for x != 0 and f(0) = k continuous at x = 0 is:

   • A.1/4✓
   • B.1/2
   • C.2
   • D.4
   Solution

   Rationalising, (sqrt(4 + x) - 2)/x = 1/(sqrt(4 + x) + 2) -> 1/4 as x -> 0. So k = 1/4.

4. Q4. The derivative of eˣ with respect to x is:

   • A.xe^(x-1)
   • B.eˣ✓
   • C.ln(x)
   • D.e^(x-1)
   Solution

   The exponential function eˣ is its own derivative: d/dx[eˣ] = eˣ.

5. Q5. If y = (1 + x)*(1 + x^2)*(1 + x^4), then dy/dx at x = 1 equals:

   • A.32
   • B.20
   • C.24
   • D.28✓
   Solution

   y = 2*2*2 = 8 at x=1; y'/y = 1/(1+x) + 2x/(1+x^2) + 4x^3/(1+x^4) = 1/2 + 1 + 2 = 7/2, so y' = 8*(7/2) = 28.

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