By Dovermann K.H.

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3. 4 10 8. 2 10 9 8 6. 10 1. 10 7 8. 10 9 4. 10 7 6. 10 7 9 4. 10 2. 10 7 2. 15: Compare 2x and x6 . 16: Compare 2x and x6 . 22 (Comparison with Polynomials). A different way of illustrating the growth of an exponential function is to compare it with the growth of a polynomial. 16 you see the graphs of an exponential function (f (x) = 2x ) and a polynomial (p(x) = x6 ) over two different intervals, [0, 23] and [0, 33]. In each figure, the graph of f is shown as a solid line, and the one of p as a dashed line.

DEFINITION OF THE DERIVATIVE 43 I and x0 belongs to I. Remember also that the domain of a function is the set on which it is defined. 2. Let f be a function and x0 an interior point of its domain. 2 We call l(x) the tangent line to the graph of f at x0 . We denote the slope of l(x) by f (x0 ) and call it the derivative of f at x0 . We also say that f (x0 ) is the slope of the graph of f at x0 and the rate of change3 of f at x0 . To differentiate a function at a point means to find its derivative at this point.

3. 2 in a less elegant but more practical way. ” Instead of asking for a line we ask for a number m, its slope, and use the line l(x) = f (x0 ) + m(x0 − x). 9. Let f be a function and x0 an interior point of its domain. 8) l(x) = f (x0 ) + m(x − x0 ). We denote its slope m by f (x0 ) and call it the derivative of f at x0 . We also say that f (x0 ) is the slope of the graph of f at x0 and the rate of change. To differentiate a function at a point means to find its derivative at this point. We provide one more reformulation which makes some calculations look more elegant.