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Ch-5: Continuity and Differentiability
12 Questions
Before You Practice Try each question on your own first. Use the Show Answer button only after you have selected your option, then read the explanation and revise the related concept from your chapter notes.
1. A function f(x) is said to be continuous at x = a if:
A. lim(x→a) f(x) = f(a)
B. f(a) is defined only, with no other condition
C. f is differentiable at a
D. f is bounded near a
Show Answer
Answer: A. lim(x→a) f(x) = f(a)
Explanation: Continuity at a point requires that the limit of the function as x approaches a equals the function's value at a.
2. Every differentiable function is necessarily:
A. Continuous
B. Discontinuous
C. Never continuous
D. Unrelated to continuity
Show Answer
Answer: A. Continuous
Explanation: Differentiability at a point implies continuity at that point, though the converse is not always true.
3. The function f(x) = |x| is:
A. Continuous everywhere but not differentiable at x = 0
B. Differentiable everywhere
C. Discontinuous at x = 0
D. Neither continuous nor differentiable anywhere
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Answer: A. Continuous everywhere but not differentiable at x = 0
Explanation: |x| has a sharp corner at x = 0, so while it is continuous there, its left-hand and right-hand derivatives differ.
4. The derivative of sin x with respect to x is:
A. cos x
B. -cos x
C. -sin x
D. sec²x
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Answer: A. cos x
Explanation: This is a standard derivative result: d/dx (sin x) = cos x.
5. Rolle's theorem requires the function to be:
A. Continuous on [a, b], differentiable on (a, b), and f(a) = f(b)
B. Only continuous on [a, b]
C. Only differentiable on (a, b)
D. Strictly monotonic on [a, b]
Show Answer
Answer: A. Continuous on [a, b], differentiable on (a, b), and f(a) = f(b)
Explanation: These are the exact hypotheses of Rolle's theorem, guaranteeing at least one point where f'(c) = 0.
6. The Mean Value Theorem states that there exists a point c in (a, b) such that:
A. f'(c) = [f(b) - f(a)] / (b - a)
B. f'(c) = 0 always
C. f(c) = 0
D. f''(c) = 0
Show Answer
Answer: A. f'(c) = [f(b) - f(a)] / (b - a)
Explanation: The Mean Value Theorem guarantees a point where the instantaneous rate of change equals the average rate of change over [a, b].
7. If y = e^x, then dy/dx equals:
A. e^x
B. x · e^(x-1)
C. e^x · ln x
D. 1
Show Answer
Answer: A. e^x
Explanation: The exponential function e^x is unique in that it is its own derivative.
8. Logarithmic differentiation is especially useful for functions of the form:
A. [f(x)]^g(x)
B. Simple polynomials
C. Linear functions
D. Constant functions
Show Answer
Answer: A. [f(x)]^g(x)
Explanation: Taking the logarithm of both sides converts a variable exponent into a product, making differentiation of such functions much simpler.
9. The derivative of the natural logarithm function log x with respect to x is:
A. 1/x
B. x
C. log x
D. 1
Show Answer
Answer: A. 1/x
Explanation: This is a standard derivative result: d/dx (ln x) = 1/x, valid for x > 0.
10. The chain rule is primarily used to differentiate:
A. Composite functions
B. Only the sum of two functions
C. Only constant functions
D. Only polynomial functions
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Answer: A. Composite functions
Explanation: The chain rule allows differentiation of a function of a function, i.e., f(g(x)), by multiplying the outer and inner derivatives.
11. The second order derivative of a function represents the:
A. Rate of change of the first derivative
B. The original function itself
C. Only the slope of the tangent line
D. The area under the curve
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Answer: A. Rate of change of the first derivative
Explanation: The second derivative measures how the rate of change (first derivative) itself is changing, often related to concavity.
12. A function can be differentiable at a point only if it is first:
A. Continuous at that point
B. Bounded at that point
C. Monotonic at that point
D. Periodic
Show Answer
Answer: A. Continuous at that point
Explanation: Continuity is a necessary (but not sufficient) condition for differentiability at a point.
