Tangent Lines

Example: Find the equation of the tangent line to the graph of y=sin[x] at the point .

Solution: The slope of the tangent line at the point is . The point-slope formula for a line gives the equations of the tangent line, . Here is the graph of y=Sin[x] plotted together with the tangent line.

     -Graphics-

Example: Show that Sin[x] is approximately equal to x for x near 0.

Solution: The tangent line approximation theorem says that f[x] is approximately equal to for x near a. On our case f[x]=Sin[x], and a=0, so Sin[x] is approximately f[0] + f'[0](x-0) = 0 + 1(x-0) = x. Here is a table of values of x and Sin[x] for x near 0 to compare.

     0            0
     
     0.01         0.00999983
     
     0.02         0.0199987
     
     0.03         0.0299955
     
     0.04         0.0399893
     
     0.05         0.0499792
     
     0.06         0.059964
     
     0.07         0.0699428
     
     0.08         0.0799147
     
     0.09         0.0898785
     
     0.1          0.0998334

     -Graphics-

Example: Find the equation of the line tangent to the curve defined by

at the point {4,2}.

Solution: Treating y[x] as an implicitly defined function of x and taking the derivative of the equation of the curve implicitly gives

                                  2
     y[x] + x y'[x] - Sin[x - y[x] ] (1 - 2 y[x] y'[x]) == 0

                                2
                   -Sin[x - y[x] ] + y[x]
     {{y'[x] -> -(-------------------------)}}
                                    2
                  x + 2 Sin[x - y[x] ] y[x]

Substuting x->4, y->2 gives the slope at this point:

The equation of the tangent line is then y = 2 - (1/2)(x-4) = 4 - x/2. We can plot the curve defined by the equation using ContourPlot[] with one contour, 9 (a contour has the form f[x]==c and we may choose the values of c).