P12 - 2 We can graphically determine the derivative of e-x...

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Project 12 MAC 2311 1. Examine the function f ( x ) = x 3 - 3 x . From precalculus, what should be the two distinct possibilities for the num- ber of turning points of any cubic (degree 3) polynomial function? In order for a smooth graph to “turn around”, what value would the slope of the curve need to be at the turning point? (Illustrate with a picture.) Is the reverse true (can a graph have that slope without turning around)? Find the values of x at which f ( x ) has horizontal tangent lines. Find the zeros ( x -intercepts) of f ( x ). Calculate the slope of f ( x ) at EACH of its zeros. How many horizontal tangent lines does the function g ( x ) = x 3 + 3 x have? Can it turn around? How many zeros does it have? Sketch the functions f ( x ) and g ( x ), incorporating the above information and labelling the points where the horizontal tangent lines occur. 1
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Based on your discoveries so far, how does the derivative of a cubic function determine the number of turning points that it has?
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Unformatted text preview: 2. We can graphically determine the derivative of e-x . Draw f ( x ) = e x and g ( x ) = e-x on the axes below. Demonstrate in your drawing that the slope of g ( x ) at x is the negative of the slope of f ( x ) at-x . That is, g ( x ) = . (Check with problem 15 on page 224.) A population of wolves increases so that the number of wolves x years from now is given by the function W ( x ) = 200(1-e-x ). Calculate W ( x ). At what rate does the population change after 1 year? 3 years? (Include units and round to the nearest tenth). Sketch the graph of W ( x ) for x ≥ 0. Label any asymptotes and intercepts. Evaluate lim x →∞ W ( x ) and lim x →∞ W ( x ). What is the interpretation of these limits in terms of the wolf population and its growth rate as time passes (to infinity). 2...
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P12 - 2 We can graphically determine the derivative of e-x...

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