# Calculus without Limits by Dovermann K.

By Dovermann K.

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Example text

We demonstrate this idea first with two examples. In the first example we use the exponential function to illustrate several ideas which enter into the general definition of the derivative. In the second example we take a geometric approach and interpret the tangent line in a special case. The geometry tells us what the derivative of the function is, or should be. 2 on page 43. The requirement that the difference between the graph and the tangent line is small is expressed analytically in terms of an inequality.

1 You are encouraged to use geometric reasoning to come up with a justification of this statement. You may also measure the angle in the figure. CHAPTER 2. 5: The graph of sin x Exercise 33. Use your graphing calculator, DfW, or any other means to get the graph of f (x) = ex near x = 1/2. You can also use one of the graphs of the exponential functions from these notes. Use the graph to estimate f (1/2), the slope of the tangent line at this point. 1 Definition of the Derivative In both of the previous examples we were able to suggest the tangent line to the graph of a function at a point.

This verifies the remaining claim in the theorem. 13 on page 22 we have the laws of logarithms. 11. The other parts are assigned as exercises below. 19 (Laws of Logarithms). For any positive real number a = 1, for all positive real numbers x and y, and any real number z loga (1) = 0 loga (a) = 1 loga (xy) = loga (x) + loga (y) loga (x/y) = loga (x) − loga (y) loga (xz ) = z loga (x) Because the exponential and logarithm functions are inverses of each other, their rules are equivalent. In the following exercises you are asked to verify this.