## The Calculus Primer (2011)

### Part IV. Using the Derivative

### Chapter 13. INSTANTANEOUS RATES OF CHANGE

**4—3. Physical Changes.** Many of the phenomena of physical science have to do with changing quantities. Thus *force, pressure, volume, temperature, work, power*, and the like are subject to change; indeed, it is not until we study such changes quantitatively that we really begin to understand the nature of our physical environment. The calculus, and particularly the derivative, furnishes the scientist, the technologist, and the engineer with a powerful tool to study phenomena involving change. The history of science affords eloquent testimony of the role played here by the calculus; for it was not until shortly after the invention of the calculus, roughly at the beginning of the eighteenth century, that modern science began to make tremendous advances in all fields, and notably in mechanics and astronomy.

We shall illustrate the application of the derivative to problems dealing with changing quantities.

EXAMPLE 1.A square metal plate, when heated, expands in area. Find the rate at which the area is increasing, per unit change in the length of the side *s*, at the instant when *s* = 6 cm.

*Solution*.*A* = *s*^{2}

= 2*s*

When *s* = 6 cm., = 12 sq. cm. per cm., or 12 sq. cm./cm.

EXAMPLE 2.The strength *M* of a certain beam varies with the thickness of the beam ft according to the formula *M* = 8.5*h*^{2}. Find the rate at which the strength increases at the instant when *h* = 5 in.

*Solution*.*M* = 8.5*h*^{2}

= 2(8.5)*h*

When *h* = 5, = 85 units/inch.

EXAMPLE 3.The electromotive force developed by a thermoelectric couple is given by *E* = 15*t* + 0.01*t*^{2}, where *t* is the temperature in degrees, and *E* is in microvolts. How fast is *E* changing when *t* = 1000?

*Solution*. = 15 + .02*t*

When *t* = 1000, = 15 + (.02)(1000) = 35 microvolts per degree.

**EXERCISE 4—1**

**1.** The area of a circular metal plate increases as the temperature rises. Find the rate at which the area is increasing, per unit change in the radius, at the instant when *r* = 10.3 in.

**2.** The volume of a spherical rubber balloon increases as the gas pressure within increases. Find the rate at which the volume is increasing at the instant when *r* = 12 in.

**3.** The moment of inertia *I* of a square beam is given by

where *h* is the thickness of the beam. Find the rate of change in *I* when *h* equals 3.

**4.** The heat developed in an electric conductor in one second is equal to

*H* = 0.24*I*^{2}*R*,

where *H* is in calories and *I* is in amperes. For a fixed resistance of 20 ohms (*R* = 20), how fast is the heat increasing at the instant when *I =* 5?

**5.** The volume (*V* cu. in.) of a certain gas varies with the pressure (*p* lb. per sq. in.) as given by

Find the rate at which *V* changes per unit change in *p*, at the instant when *p* = 20. Is the volume increasing or decreasing?

**6.** The theoretical discharge when water flows over a weir or a dam is

*Q* = 3.33*lh*^{3/2},

where *Q* is the amount of water in cu. ft. per sec., *l* = the width of the weir in feet, and *h* = the height of the water in feet (the head) above the weir. For a weir 200 feet wide, find the rate at which the amount of flow is increasing at the instant when *h* = 9 ft.

**7.** The heat of absorption of an ammonia solution in a refrigerating system is

*Q* = 887 − 350*x* − 400*x*^{2},

where *Q* is the amount of heat in B.T.U., and *x* is the concentration of the ammonia solution. How fast is *Q* changing at the instant when *x* = .18?

**8.** The power required to propel a ship of a certain design is given by

where *H* = horsepower, *d* = the ship’s displacement in long tons, and *v* = the ship’s speed in knots. For a ship with a displacement of 8000 long tons, how much power is required to propel the ship at 20 knots? How fast is the required power increasing at the instant when *v* = 20 knots?

**9.** The quantity of heat radiated by a surface varies directly as the fourth power of its absolute temperature:

where *Q* is the amount of heat in B.T.U.; *A* = the area of the surface in sq. ft.; *z* = the time in hours; *T* = the absolute temperature; and *k* is a constant depending upon the particular surface. Find how fast *Q* is changing when *T* = 2000° for one square foot of surface for one hour; take *k* = .15.

**10.** The velocity with which a gas flows through a small hole varies inversely as the square root of the density *d* of the gas. (a) If *V* = 50 cu. in. per hr. for a gas whose density *d* = .0225, find the formula for the velocity of flow, (b) Find the rate at which *V* is changing at the instant when *d* = .01.