1. If a function is differentiable at a point, it must be:
a) Continuous at that point
b) Discontinuous at that point
c) Neither continuous nor discontinuous
d) None of the above
Answer:
a) Continuous at that point
Explanation:
Differentiability at a point implies continuity at that point.
2. Which of the following functions is not continuous at x = 0?
a) sin(x)
b) cos(x)
c) tan(x)
d) x^2
Answer:
c) tan(x)
Explanation:
tan(x) is undefined at x = π/2 and its multiples, including x = 0.
3. The derivative of a constant is:
a) 0
b) 1
c) The constant itself
d) Undefined
Answer:
a) 0
Explanation:
Constants don't change, so their rate of change (derivative) is zero.
4. If f(x) = |x|, then f is:
a) Continuous everywhere but not differentiable at x = 0
b) Differentiable everywhere
c) Neither continuous nor differentiable at x = 0
d) Discontinuous at x = 0
Answer:
a) Continuous everywhere but not differentiable at x = 0
Explanation:
The modulus function has a sharp turn at x = 0, making it non-differentiable there.
5. The derivative of f(x) = x^3 with respect to x is:
a) 3x^2
b) x^2
c) 6x
d) 3x
Answer:
a) 3x^2
Explanation:
Using the power rule for differentiation.
6. A function which is continuous in its domain:
a) Must be differentiable
b) Must not be differentiable
c) Can be differentiable or not
d) Is always increasing
Answer:
c) Can be differentiable or not
Explanation:
Continuity doesn't guarantee differentiability.
7. The derivative of tan(x) is:
a) cos^2(x)
b) sin^2(x)
c) sec^2(x)
d) csc^2(x)
Answer:
c) sec^2(x)
Explanation:
Basic differentiation rule for tan(x).
8. If f(x) = ln(x), its derivative is:
a) x
b) 1/x
c) e^x
d) x^2
Answer:
b) 1/x
Explanation:
Basic differentiation rule for natural logarithm.
9. A function is said to be differentiable at x = a if:
a) f is defined at x = a
b) f is continuous at x = a
c) The left-hand and right-hand derivatives at x = a exist and are equal
d) f has a maximum or minimum at x = a
Answer:
c) The left-hand and right-hand derivatives at x = a exist and are equal
Explanation:
Differentiability requires the function's slope from the left and right to be the same at the point.
10. If g(x) = e^x, its derivative is:
a) e^x
b) xe^x
c) 1/e^x
d) x^2e^x
Answer:
a) e^x
Explanation:
The exponential function is its own derivative.
11. The point where the function changes its nature from increasing to decreasing or vice-versa is:
a) Point of inflection
b) Point of discontinuity
c) Critical point
d) None of the above
Answer:
c) Critical point
Explanation:
At critical points, the derivative is either zero or does not exist.
12. If a function is differentiable in (a, b), then it is:
a) Continuous in [a, b]
b) Discontinuous at a or b
c) Not necessarily continuous in [a, b]
d) None of the above
Answer:
a) Continuous in [a, b]
Explanation:
Differentiability implies continuity, but not vice-versa.
13. The derivative of sin^2(x) with respect to x is:
a) 2sin(x)cos(x)
b) 2cos(x)sin(x)
c) 2sin^2(x)
d) 2cos^2(x)
Answer:
a) 2sin(x)cos(x)
Explanation:
Using the chain rule and the derivative of sin(x).
14. If h(x) = 1/x, h is not differentiable at:
a) x = 1
b) x = 0
c) x = -1
d) x = 2
Answer:
b) x = 0
Explanation:
1/x is undefined at x = 0.
15. The second derivative measures:
a) Slope of the tangent
b) Rate of change of the function
c) Curvature of the function
d) None of the above
Answer:
c) Curvature of the function
Explanation:
The second derivative gives insight into the concavity or convexity of a function.
16. The value of d/dx [x^x] at x = 1 is:
a) 1
b) 2
c) 0
d) e
Answer:
b) 2
Explanation:
Differentiating x^x and evaluating at x = 1 gives the result.
17. The derivative of f(x) = log(x) to the base a (where a > 0, a ≠ 1) is:
a) 1/(x ln(a))
b) ln(a)/x
c) 1/x
d) a^x
Answer:
a) 1/(x ln(a))
Explanation:
Differentiation rule for logarithms with bases other than e.
18. A function which is differentiable on its domain:
a) Will have a sharp corner in its graph
b) Will have breaks in its graph
c) Will have a smooth curve without any sharp turns
d) None of the above
Answer:
c) Will have a smooth curve without any sharp turns
Explanation:
Differentiability ensures a smooth curve.
19. The derivative of the function f(x) = √x is:
a) 1/2√x
b) 2√x
c) x^(-1/2)
d) x^(3/2)
Answer:
a) 1/2√x
Explanation:
Using the power rule for differentiation.
20. If y = u/v and both u and v are differentiable functions of x, then dy/dx is:
a) (v du/dx – u dv/dx)/v^2
b) (u du/dx – v dv/dx)/v^2
c) (u dv/dx + v du/dx)/v^2
d) (u dv/dx – v du/dx)/v^2
Answer:
a) (v du/dx – u dv/dx)/v^2
Explanation:
Using the quotient rule for differentiation.