2.2   DIFFERENTIALS  AND  TANGENT  LINES

Suppose we are given a curve y=f(x) and at a point (a, b) on the curve the slope f '(a) is defined. Then the tangent line to the curve at the point (a, b), illustrated in Figure 2.2.1, is defined to be the straight line which passes through the point (a, b) and has the same slope as the curve at x=a. Thus the tangent line is given by the equation

l (x) - b =f '(a) (x-a),

Or                        l (x)   =f '(a) (x-a)+ b.

Figure 2.2.1  Tangent lines

EXAMPLE 1 

For the curve y=x3, find the tangent lines at the points (0, 0), (1,1), and (___, ____)

(Figure 2.2.2). The slope is given by f ′(x) = 3x². At x= 0, f ′(0)=3·0² = 0. The tangent line has the equation

y=0(x-0) + 0,  or  y=0.

Figure 2.2.2

             At x = 1, f ′(1) = 3, whence the tangent line is

y=3(x-1) +1,   or   y=3x-2.

             At x = ____, f '(_____) = 3 ·(___)² =_____, so the tangent line is

Given a curve y= f(x), suppose that x starts out with the value a and then changes by an infinitesimal amount Δx. What happens to y? Along the curve, y will change by the amount

f(a +Δ x) - f(a) =Δ y.

But along the tangent line y will change by the amount

l(a + Δx) - l (a) = [ f ′(a)(a+ Δx-a) + b] -[ f ′(a) (a- a) +b]

                       =f ′(a) Δx.

When x changes from a to a + Δx, we see that:

                      change in y along curve = f(a+ Δx) - f(a),

                  change in y along tangent line = f ′(a) Δx.

In the last section we introduced the dependent variable Δy, the increment of y, with the equation

Δy= f(x + Δx) - f(x).

Δy is equal to the change in y along the curve as x changes to x + Δx.

The following theorem gives a simple but useful formula for the increment Δy.

INCREMENT  THEOREM

              Let y=f(x). Suppose f ′(x) exists at a certain point x, and Δx is infinitesimal.

              Then Δy is infinitesimal, and

Δy= f ′(x) Δx + ε Δx

for some infinitesimal ε, which depends on x and Δx.

PROOF

Case 1   Δx=0. In this case, Δy=f ′(x) Δx = 0, and we put ε=0.

Case 2   Δx≠ 0 . Then

        So for some infinitesimal ε,

Multiplying both sides by Δx,

Δy= f ′(x)Δ x + ε Δx.

EXAMPLE  2  Let y=x3, so that y′=3x2. According to the Increment Theorem,

Δy= 3x2 Δx +ε Δx

       for some infinitesimal ε . Find ε in terms of x and Δx when Δx≠ 0. We have

       We must still eliminate Δy. From Example 1 in Section 2.1,

Substituting,              ε = (3x²+ 3xΔx + (Δx) ²) - 3x².

Since 3x² cancels,          

ε = 3 xΔx+ (Δx) ².

We shall now introduce a new dependent variable dy, called the differential of y, with the equation

dy = f ′(x) Δx.

dy is equal to the change in y along the tangent line as x changes to x+ Δx. In Figure 2.2.3 we see dy and Δy under the microscope.

Figure 2.2.3

To keep our notation uniform we also introduce the symbol dx as another name for Δx. For an independent variable x, Δx and dx are the same, but for a dependent variable y. Δy and dy are different.

DEFINITION

         Suppose y depends on x, y=f(x).

         ( i) The differential of x is the independent variable dx = Δx.

         (ii) The differential of y is the dependent variable dy given by

dy = f ′(x)dx.

          When dx ≠ 0, the equation above may be rewritten as

Compare this equation with

The quotient dy/ dx is a very convenient alternative symbol for the derivative f ′(x). In fact we shall write the derivative in the form dy/ dx most of the time.

The differential dy depends on two independent variables x and dx. In functional notation,

  dy= df(x, dx)

Where df is the real function of two variables defined by

df(x, dx) = f ′(x)dx.

When dx is substituted for Δx and dy for f ′(x)dx, the Increment Theorem takes the short form

Δy= dy + ε dx.

The Increment Theorem can be explained graphically using an infinitesimal microscope. Under an infinitesimal microscope, a line of length Δx is magnified to a line of unit length, but a line of length ε Δx is only magnified to an infinitesimal length ε. Thus the Increment Theorem shows that when f ′(x) exists:

  (1) The differential dy and the increment Δy=dy+ε dx are so close to each other that they

     cannot be distinguished under an infinitesimal microscope.

  (2) the curve y=f(x) and the tangent line at (x,y) are so close to each other that they cannot

     be distinguished under an infinitesimal microscope; both look like a straight line of

     slope f ′(x).

Figure 2.2.3 is not really accurate. The curvature had to be exaggerated in order to distinguish the curve and tangent line under the microscope. To give an accurate picture, we need a more complicated figure like Figure 2.2.4, which has a second infinitesimal microscope trained on the point (a+ Δx, b+Δy) in the field of view of the original microscope. This second microscope magnifies ε dx to a unit length and magnifies Δx to an infinite length.

Figure 2.2.4

EXAMPLE 3  Whenever a derivative f ′(x) is known, we can find the differential dy at once by simply multiplying the derivative by dx, using the formula dy = f ′(x) dx. The examples in the last section give the following differentials.

        (a)  y=x³,      dy = 3x² dx.

        (b)  y=_____,   dy = _________ where x >0.

        (c)  y=1/x,      dy= -dx/x2²     when x ≠ 0.

        (d)  y =| x|,      dy= 

  (e)    y=bt - 16t², dy= (b - 32 t) dt.

The differential notation may also be used when we are given a system of formulas in which two or more dependent variables depend on an independent variable. For exmple if y and z are functions of x,

                     y= f(x),   z = g(x),

Then Δy,Δ z, dy , dz are determined by

Δy= f(x+Δx) - f(x),     Δz = g(x+ Δx) - g(x),

                     dy = f ′(x)dx,        dz= g ′(x) dx.

EXAMPLE  4  Given y=___ x, z=x³, with x as the independent variable , then

The meaning of the symbols for increment and differential in this example will be different if we take y as the independent variable. Then x and z are functions of y.

x=2y,     z=8y3.

Now Δy=dy is just an independent variable, while

Moreover,                dx=2dy, dz=24y² dy.

We may also apply the differential notation to terms. If l(x) is a term with the variable x, then l(x) determines a function f,

  l(x) = f(x),

And the differential d( l(x)) has the meaning

                            d( l(x)) = f '(x)dx.

EXAMPLE 5

  (a)     d(x³3) = 3x² dx.

    (b)

       (c)

       (d)

       (e)    Let u= bt and w= -16 t². Then

u+ w= bt -16t²,  d(u+w) = (b-32t) dt.

PROBLEMS  FOR  SECTION 2.2

In Problems 1-8, express Δy and dy as functions of x and Δx, and for Δx infinitesimal find an infinitesimal ε such that Δy=dy+ ε Δ x.

1   y=x2                             2   y= -5x2

3   ________                 4  y=x4 

5   y=1/x                    6    y=x -2

7   y=x -1/x  8    y=4x + x3

9   If y = 2x² and z =x³ , find Δy, Δz, dy, and dz.

10  If y = 1/(x+ 1) and z =1/(x+ 2), find Δy, Δz, dy, and dz.

11  Find d(2x+1)              12  Find d(x² - 3x)

13  Find d(_______)           14  Find d(_______)

15  Find d(ax+b)              16  Find d(ax²)

17  Find d(3+2/x)              18  Find d(_______) 

19  Find d(_______)            20  Find d(x³ - x²)

21  Let y=_____, z=3x. Find d(y+z) and d(y/z).

22  Let y= x -1, z=x³. Find d(y+z) and d(yz).

In Problems 23-30 below, find the equation of the line tangent to the given curve at the given point.

23  y=x²; (2, 4)             24   y=2x²; (-1, 2)  

25  y= -x²; (0, 0)            26   _____ (1, 1)

27  y=3x - 4; (1, -1)          28   ________(5, 2)       

29  y=x 4; ( -2, 16)           30    y=x 3 - x;   ( 0, 0)

31  Find the equation of the line tangent to the parabola y = x² at the point (x0, ____).

32  Find all points P (x0, ____) on the parabola y = x² such that the tangent line at P passes

    through the point (0, -4)

□33 Prove that the line tangent to the parabola y=x² at P(x0, ____ ) does not meet the

     parabola at any point except P.