121616949-math.38 - 24 Chapter 2 Instantaneous Rate Of...

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24 Chapter 2 Instantaneous Rate Of Change: The Derivative 6. Find an algebraic expression for the difference quotient ( f ( x + Δ x ) - f ( x )) / Δ x when f ( x ) = mx + b . Simplify the expression as much as possible. Then determine what happens as Δ x approaches 0. That value is f ( x ). 7. Sketch the unit circle. Discuss the behavior of the slope of the tangent line at various angles around the circle. Which trigonometric function gives the slope of the tangent line at an angle θ ? Why? Hint: think in terms of ratios of sides of triangles. 8. Sketch the parabola y = x 2 . For what values of x on the parabola is the slope of the tangent line positive? Negative? What do you notice about the graph at the point(s) where the sign of the slope changes from positive to negative and vice versa? We started the last section by saying, “It is often necessary to know how sensitive the value of y is to small changes in x .” We have seen one purely mathematical example of this: finding the “steepness” of a curve at a point is precisely this problem. Here is a more
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