The Calculus Primer (2011)
Part VII. Differentials
Chapter 28. APPROXIMATE CALCULATIONS
7—8.The Differential as an Approximation. In §7—4 we learned that for a very small change in the independent variable, the differential of the function is very nearly equal to the increment of the function; or, dy = Δy,approximately. This relationship is of practical value in determining the approximate change in a function corresponding to a particular value of the independent variable together with a small change in that variable.
EXAMPLE 1.What is the change in the area (y) of a square when the side of the square is increased from 8.0 to 8.2 inches?
y = x2,
When x = 8, = 16. Let the increase in x be Δx; thus Δx = .2. Let us assume that Δy = dy; then, since hence Δy = (16) (.2) = 3.2.
The value of y when x = 8.2 is equal to 67.24; hence the exact value of the increment in y equals 67.24 − 64.00 = 3.24, which is very close to the approximate value Δy = 3.2 found above.
NOTE 1. The reader should study the diagram and note the relation of the approximate formula Δy = 2x·Δx with the exact formula Δy = 2x·Δx + (Δx)2.
NOTE 2. The magnitude of the error introduced by using the value of the differential in lieu of the increment of the function will depend, in general, upon (1) the value taken for the independent variable, (2) the size of the increment given to the independent variable, and (3) the nature of the function.
EXAMPLE 2.In the function y = x8, by how much will y increase when x changes from 3 to 3.02?
NOTE. To compute the exact value of Δy, or, for that matter, the exact value of (3.02)8, involves considerable labor. By five-place logarithms, (3.02)8 = 6919.6; hence the exact value of
Δy = 6919.6 − (3)8 = 6919.6 − 6561 = 358.4,
as compared with the approximate value 350 found above.
EXAMPLE 3.The edge of a cubical block of metal is equal to 9.7 cm., but was measured as 10 cm. About how great is the error in the computed volume due to the error in measuring? What is the per cent of error in the volume?
V = x3,
when x = 10, = 300.
Hence ΔV = Δx = (300) (.3) = 90 cu. cm., approx.
The per cent of error in the measurement of the edge is , or 3%; the per cent of error in the volume is about , or about 9%.
NOTE. The reader should note the interpretation, in the diagram, of the expression
V + ΔV = x3 + 3x2·Δx + 3x(Δx)2 + (Δx)3;
in this way he will see the significance of the approximation formula ΔV = 3x2(Δx).
EXAMPLE 4.Find, approximately, the square root of 402.
Solution. Let y = .
Here x may be taken as 400, and Δx = 2.
therefore = + Δy = 20 + .05 = 20.05, approx.
1. A metal cube when heated expands so that its edge increases from 20 cm. to 20.003 cm. Find (a) the approximate change in its area; (b) the approximate change in its volume.
2. Find the approximate change both in the circumference and in the area of a circle when the radius is decreased from 5 inches to 4.98 inches.
3. If the diameter of a sphere is actually 19.98 in., but is measured as 20.02 in., find the approximate error in computing both the surface and the volume of the sphere.
4. Find the change in the volume of a cylindrical metal bearing of length 10 cm. when it wears down from a radius of 2 cm. to a radius of 1.99 cm.
5. Find, by differentiation, the approximate square root of 291.
6. Find, by differentiation, the approximate value of
7. Approximately what error would be allowable in the side of a square, about 40 cm. on a side, if the error in the calculated area is not to exceed 2 sq. cm.?
8. The pressure p in lb. per sq. in. of a gas in a closed container at constant temperature varies according to the law pv = k. Derive (a) an expression for the approximate change in pressure if the volume is changed by a small amount; (b) an expression for the approximate change in volume if the pressure is changed by a small amount.
9. The inside diameter of a circular concrete conduit is 18 in.; the shell is in. thick. Find the approximate area of the cross-section of the concrete, i.e., the area of the “ring” between the inside and outside of the shell.
10. The heat generated by an electric current is given by the formula H = 0.24I2Rt, where H is in calories, I = amperes, R is the resistance in ohms, and t = the time in seconds. Find the increase in the amount of heat generated per second by a constant resistance of 50 ohms when I changes from 20 to 22 amperes.
11. The horsepower required to propel a ship of a certain design is where D is the displacement in long tons and v the speed in knots. Find the additional power required to increase the speed of a 27,000 ton vessel from 25 to 27 knots.
12. Prove that the change in the circumference of a circle caused by a change in the radius is independent of the original circumference and radius. If a blacksmith inserted a piece of iron one inch long in the metal tire of a wagon wheel originally 42 in. in diameter, how much “space” would there be between the rim of the wheel and the tire? If a piece of metal 1 in. long were inserted in a metal bracelet which originally fitted the arm snugly, by how much would it stand away after lengthening?
1. Given the function x½ + y½ = a½, find ; also, .
2. Find :
(a) x2 + y2 + 2x = 5y
(b) (x2 − y2)2 = x2 + y2
3. (a) Find the slope of the curve y = x(x2 − 1) at the point of inflection.
(b) Find the slope of the tangent to the curve x2 − xy + y2 = 6 at the point (−2,1).
4. The distance S traveled by a body projected vertically upwards in time t is given by the equation S = 192t − 16t2. Find the greatest height the body will reach, and how long it will take to reach it.
(c) log (px2 + qx + r)
6. Find for each of the following:
(a) b2x2 + a2y2 = a2b2
(b) x2 + y2 = 2x
(c) x2y2 = x2 + y2