Quiz for Applications of Derivatives

Q1: What is the derivative of a function used to determine?

A. Slope of the curve B. Area under the curve C. Volume of a solid D. Length of a curve

Q2: A function f(x) has a minimum at x = c if:

A. f'(c) = 0 and f''(c) > 0 B. f'(c) = 0 and f''(c) < 0 C. f'(c) > 0 and f''(c) < 0 D. f'(c) < 0 and f''(c) > 0

Q3: The point at which the tangent to a curve is horizontal:

A. f'(x) = 0 B. f'(x) = 1 C. f''(x) = 0 D. f'(x) = ∞

Q4: If a function is increasing in an interval, then:

A. f'(x) > 0 B. f'(x) < 0 C. f'(x) = 0 D. f'(x) is undefined

Q5: What does a point of inflection represent?

A. Change in concavity B. Maximum value C. Minimum value D. Undefined slope

Q6: If f'(x) = 0 and f''(x) < 0 at x = c, then:

A. f(x) has a maximum at x = c B. f(x) has a minimum at x = c C. f(x) is increasing at x = c D. f(x) is decreasing at x = c

Q7: The rate of change of area of a circle with respect to radius r is:

A. 2πr B. πr² C. π D. 2π

Q8: A particle moves such that s = t³ – 6t² + 9t. Find time when velocity is zero.

A. t = 1 and 3 B. t = 0 and 2 C. t = 3 and 4 D. t = 1 and 2

Q9: If f(x) = x³ – 6x² + 9x, then local minimum occurs at:

A. x = 3 B. x = 1 C. x = 0 D. x = 2

Q10: The maximum value of sin x + cos x is:

A. √2 B. 1 C. 2 D. 1.5

Q11: The least value of x² + 1/x² for x > 0 is:

A. 2 B. 4 C. 3 D. 5

Q12: What is the angle of intersection of two curves when their tangents are perpendicular?

A. 90° B. 45° C. 60° D. 0°

Q13: If y = f(x) is increasing, then which is true?

A. f'(x) > 0 B. f'(x) < 0 C. f'(x) = 0 D. f''(x) = 0

Q14: Find the critical point of f(x) = x⁴ – 4x³

A. x = 0, 3 B. x = 1, 4 C. x = –1, 2 D. x = 2, 3

Q15: The minimum value of (x + 1/x) for x > 0 is:

A. 2 B. 1 C. 3 D. 0

Q16: For maximum value, the second derivative should be:

A. Negative B. Positive C. Zero D. Undefined

Q17: Find point on y = x² closest to (0,1)

A. (0.5, 0.25) B. (1, 1) C. (0, 0) D. (1, 0)

Q18: A cone is melting such that height remains constant. Rate of volume decrease is:

A. πr² dr/dt B. (1/3)πr² dh/dt C. (1/3)πh² dr/dt D. πr dh/dt

Q19: What is the point of local maxima of f(x) = –x² + 4x?

A. x = 2 B. x = –2 C. x = 0 D. x = 1

Q20: In which interval is f(x) = x³ – 3x² + 2 increasing?

A. (2, ∞) B. (–∞, 0) C. (0, 1) D. (–1, 0)

Q21: The function f(x) = x³ has:

A. No local maxima or minima B. Only maxima C. Only minima D. Both maxima and minima