Application of Derivatives quiz

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is:

The function f(x) = x^2 is strictly decreasing on:

The function f(x) = 7x - 3 is:

The critical point of f(x) = x^2 - 4x + 5 is:

The minimum value of f(x) = x^2 + 2x + 3 is:

If f'(k) = 0 and f''(k) < 0, then x = k is a point of:

The maximum value of sin x cos x is:

f(x) = log x is strictly increasing on:

The function f(x) = x^3 has at x = 0:

Two positive numbers have sum 16. Their product is maximum when the numbers are:

Find the rate of change of the volume of a sphere with respect to its radius when r = 2 cm. (2 marks)

Show that f(x) = cos x is strictly decreasing on (0, pi). (2 marks)

Find the intervals in which f(x) = x^2 - 4x + 6 is increasing. (2 marks)

Find the local minimum value of f(x) = x^2 - 6x + 10. (2 marks)

Show that f(x) = x^3 - 3x^2 + 3x - 100 is increasing on R. (2 marks)

Find the maximum value of f(x) = -(x - 1)^2 + 10. (2 marks)

The side of a square increases at 4 cm/min. How fast is the area increasing when the side is 8 cm? (2 marks)

Find the critical points of f(x) = x^4 - 2x^2. (2 marks)

Show that f(x) = 2x + sin x is strictly increasing on R. (2 marks)

Prove that f(x) = log(sin x) is strictly increasing on (0, pi/2). (2 marks)

The total cost C(x) in rupees of producing x units is C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000. Find the marginal cost when 3 units are produced. (3 marks)

The volume of a cube is increasing at 9 cm^3/s. How fast is its surface area increasing when the edge is 10 cm? (3 marks)

Find the intervals of increase and decrease and the minimum value of f(x) = x^2 - 4x + 6. (3 marks)

Find the absolute maximum value of f(x) = sin x + cos x on [0, pi/2] and the point where it occurs. (3 marks)

Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. Find this minimum sum. (3 marks)

Read the passage and answer the questions. A 5 m long ladder leans against a vertical wall. A painter pulls the bottom of the ladder away from the wall along the ground at 2 cm/s. (i) Write the relation between x (distance of foot) and y (height of top). (ii) Find the rate at which the height on the wall decreases when the foot is 4 m from the wall. (iii) Is the top moving faster or slower than the foot at this instant? (5 marks)

Read the passage and answer the questions. The profit of a small firm in rupees from selling x items is P(x) = -5x^2 + 125x + 37500. (i) Find P'(x). (ii) Find the value of x for maximum profit. (iii) Find the maximum profit. (5 marks)

Read the passage and answer the questions. A farmer has 200 m of fencing to enclose a rectangular field along a straight river; no fence is needed along the river. (i) Express the area A in terms of the width x. (ii) Find x for maximum area. (iii) Find the maximum area. (5 marks)

Read the passage and answer the questions. The height of a roller coaster track over a horizontal distance x is modelled by f(x) = 4x^3 - 6x^2 - 72x + 30. (i) Find the intervals where f is increasing. (ii) Find the local maximum value. (iii) Find the local minimum value. (5 marks)

Read the passage and answer the questions. An oil spill spreads as a circle on the sea. Its radius increases at 2 m/min. (i) Find the rate of increase of area when r = 50 m. (ii) Find the rate of increase of circumference. (iii) Does the rate of change of area stay constant? Explain. (5 marks)

Find the local maxima and minima of f(x) = 3x^4 + 4x^3 - 12x^2 + 12 and the local maximum and minimum values, with a rough sketch. (6 marks)

An open tank with a square base and vertical sides is to hold 32 m^3 of water. Find the dimensions that require the least material. (6 marks)

Show that the semi-vertical angle of a right circular cone of given slant height and maximum volume is tan^-1(sqrt(2)). (6 marks)

A wire of length 28 m is cut into two pieces. One piece is bent into a square and the other into a circle. Find the lengths of the two pieces so that the combined area is minimum. (6 marks)

Find the absolute maximum and minimum values of f(x) = 4x - x^2/2 on [-2, 4.5], and find the intervals of increase and decrease with a sketch. (6 marks)