To find the line y = L(x) that best approximates a curve y = f (x) near x = c, we calcul...

To find the line y = L(x) that best approximates a curve y = f (x) near x = c, we calculate the difference between them

D(x) = f(x) - L(x)

and show that minimizing it near and at the point (c, f (c)) leads to the tangent line.

a. Show that if the difference D(x) is zero at x = c, then f(c) = L(c).

b. Use the result of part (a) to show that the line passes through the point (c, f (c)), and use the point-slope form for the equation of a line (see page 9) to show that the line has the form

L(x) = m(x - c) + f(c)

c. Substitute this expression for L(x) into the formula for D(x) to show that if D(c) = 0, then D(x) = f(x) 2 m(x - c) - f (c)

d. Since the best linear approximation will be the line for which D(x) changes from zero as little as possible near x = c, the derivative of D(x) should be zero at c. Find D'(c) and then find the m that makes = 0.

e. Find L(x) for this best slope and show that the line y = L(x) is the tangent line to the curve y = f(x) at x = c.

f. Explain why this shows that the tangent line to a curve at a point is the best linear approximation to the curve at that point.