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Explanation: Derivatives fundamentally measure the instantaneous rate of change of one quantity with respect to another.
2. A function is said to be increasing on an interval if its derivative there is:
A. Positive (≥ 0) throughout the interval
B. Negative throughout the interval
C. Zero throughout the interval
D. Undefined throughout
Answer: A. Positive (≥ 0) throughout the interval
Explanation: A non-negative derivative throughout an interval indicates the function's values are non-decreasing there.
3. A point where f'(x) = 0 is called a:
A. Critical point
B. Point of inflection, always
C. Point of discontinuity
D. Asymptote
Answer: A. Critical point
Explanation: Critical points occur where the derivative is zero (or undefined) and are candidates for local maxima or minima.
4. The second derivative test uses f''(x) mainly to determine:
A. Whether a critical point is a local maximum or minimum
B. Whether the function is continuous
C. The domain of the function
D. The range of the function
Answer: A. Whether a critical point is a local maximum or minimum
Explanation: A positive f''(x) at a critical point indicates a local minimum, while a negative value indicates a local maximum.
5. If f''(x) > 0 at a critical point, that point is a:
A. Local minimum
B. Local maximum
C. Point of inflection
D. Point of discontinuity
Answer: A. Local minimum
Explanation: A positive second derivative indicates the curve is concave upward at that point, corresponding to a local minimum.
6. The equation of the tangent to the curve y = f(x) at the point (x1, y1) is:
A. y - y1 = f'(x1)(x - x1)
B. y = f'(x1)
C. x - x1 = f(x1)
D. y + y1 = f'(x1)
Answer: A. y - y1 = f'(x1)(x - x1)
Explanation: This is the standard point-slope form of the tangent line, where f'(x1) is the slope of the curve at that point.
7. Rate of change problems (such as related rates) are typically solved using:
A. Derivatives with respect to time
B. Integration with respect to time
C. Purely algebraic equations
D. Matrices
Answer: A. Derivatives with respect to time
Explanation: Since rate of change refers to how a quantity varies with time, such problems are naturally modeled using derivatives with respect to t.
8. The normal to a curve at a given point is:
A. Perpendicular to the tangent at that point
B. Parallel to the tangent at that point
C. The same line as the tangent
D. Always horizontal
Answer: A. Perpendicular to the tangent at that point
Explanation: By definition, the normal line is drawn perpendicular to the tangent line at the point of contact.
9. Maxima and minima problems in real-world optimization are generally solved using the:
A. First and second derivative tests
B. Method of integration only
C. Concept of limits only
D. Matrices and determinants
Answer: A. First and second derivative tests
Explanation: Optimization problems are solved by finding critical points using the first derivative and classifying them using the second derivative test.
10. A function is said to be decreasing on an interval if its derivative there is:
A. Negative (≤ 0) throughout the interval
B. Positive throughout the interval
C. Zero throughout the interval
D. Undefined throughout
Answer: A. Negative (≤ 0) throughout the interval
Explanation: A non-positive derivative throughout an interval means the function's values are non-increasing there.
11. The approximate value of a function near a known point can be estimated using:
A. Differentials
B. Definite integrals
C. Determinants
D. Matrices
Answer: A. Differentials
Explanation: The concept of differentials, dy = f'(x)dx, provides a linear approximation for small changes in a function.
12. The absolute maximum of a continuous function on a closed interval occurs at:
A. A critical point or one of the endpoints of the interval
B. Only at critical points inside the interval
C. Only at the endpoints
D. Nowhere on a closed interval
Answer: A. A critical point or one of the endpoints of the interval
Explanation: The Extreme Value Theorem guarantees that the absolute maximum on a closed interval is attained either at a critical point or at an endpoint.
This page contains a chapter-wise objective practice set. Try the questions without checking the answer first, then use the explanation to understand the point you missed. The current page contains 12 questions, so you can practise in smaller sets when you are revising.
Quick revision tip
After the practice set, write down the question numbers you missed and revise only those concepts once more. This makes MCQ practice more useful than repeatedly attempting the same questions without reviewing mistakes.
