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Extra resources for Applied calculus

Example text

3) f (x) = − x −x . =√ y 1 − x2 This is exactly the result predicted in the beginning of the discussion. 61) when we formally calculate the derivative of this specific function. Exercise 32. 5 you see part of the graph of the function f (x) = sin x. In this picture draw a line to resemble the graph near the point (1, sin 1). Determine the slope of the line which you drew. Write out the equation for this line in point slope form. Find f (1). 1 You are encouraged to use geometric reasoning to come up with a justification of this statement.

You assume that the population grows exponentially. At time t = 0 you start out with a population of 800 bacteria. After three hours the population is 1900. What is the relative growth rate for the population? How long did it take for the population to double. How long does it take until the population has increased by a factor 4? Remark 4. Some problems remain unresolved in this section. 12. 19: loga (xy) = loga (x) + loga (y) and loga (xz ) = z loga (x), and we have to define the Euler number e.

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. In the first figure you see that, on the given interval, the polynomial p is substantially larger than the exponential function f . In the second figure you see how the exponential function has overtaken the polynomial and begins to grow a lot faster.