Applications of Derivatives — Practice

Question 1

Find the rate of change of the area of a circle with respect to its radius when r=5 cm.

Accepted Answer

A=pi r^2, dA/dr=2 pi r. At r=5: dA/dr=10 pi. Answer: 10 pi cm^2/cm.

Question 2

Show that f(x)=7x-3 is increasing on R.

Accepted Answer

f'(x)=7>0 for all x. Answer: Since f'(x)>0 everywhere, f is strictly increasing on all of R.

Question 3

The side of a square is increasing at the rate of 3 cm/s. Find the rate of increase of its area when the side is 10 cm.

Accepted Answer

A=x^2, dA/dt=2x.dx/dt. At x=10, dx/dt=3: dA/dt=2(10)(3)=60. Answer: 60 cm^2/s.

Question 4

Find the local maxima and minima of f(x)=x^3-3x.

Accepted Answer

f'(x)=3x^2-3=3(x-1)(x+1), zero at x=1,-1. f' is + for x<-1, - for -1<x<1, + for x>1. Answer: Local maximum at x=-1 (value 2); local minimum at x=1 (value -2).

Question 5

Use the second derivative test to classify the critical point of f(x)=x^2-4x+3.

Accepted Answer

f'(x)=2x-4, zero at x=2. f''(x)=2>0. Answer: Since f''(2)>0, x=2 is a local minimum.

Question 6

Find the absolute maximum and minimum of f(x)=x^2 on [-3,1].

Accepted Answer

f'(x)=2x=0 at x=0 (in range), f(0)=0. Endpoints: f(-3)=9, f(1)=1. Answer: Absolute maximum is 9 at x=-3; absolute minimum is 0 at x=0.

Question 7

Find two positive numbers whose sum is 20 and whose product is maximum.

Accepted Answer

Let the numbers be x and 20-x. P=x(20-x)=20x-x^2. P'=20-2x=0 gives x=10. P''=-2<0, confirming a maximum. Answer: The two numbers are 10 and 10.

Question 8

The total cost function is C(x)=0.005x^3-0.02x^2+30x+5000. Find the marginal cost when x=10 units are produced.

Accepted Answer

MC=dC/dx=0.015x^2-0.04x+30. At x=10: MC=0.015(100)-0.04(10)+30=1.5-0.4+30=31.1. Answer: Marginal cost is Rs 31.10 (approximately).

Practice — Applications of Derivatives

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