The Derivative Function

1. The Derivative Function

2. Bacteria Growth • From looking at the the average rate of change of this function from data collected • every two hours determine: • Where is the function increasing/decreasing? • Where is the function concave up/down?

3. Warming UP Exercise 7 from Derivative at a Point Consider the graph below. The domain of the function is all the real numbers. Assume that outside the window the function continues the same behavior as the one indicated in the window. Where is f(x) increasing? Where is f(x)>0? Where is f(x) concave up?

4. i. Drawthe tangent line at each of the given points and use the grid to complete the table below. All the answers are estimates

5. ii. Use interval notation to complete the following information a. Intervals where the derivative is negative (solutions to f ‘ (x) < 0) b. Intervals where f(x) < 0. Describe those points graphically. c. Intervals where the derivative is positive (solutions to f ‘ (x) > 0) d. Intervals where f (x) > 0. Describe those points graphically.

6. Critical Points of A Continuous Function A critical points of a continuous function y=f(x) is a point in its domainwhere either f‘(x)=0or f ‘(x) is undefined. f’(x)=0 when the tangent line is horizontal f’(x) is undefined at a point in the domain where the tangent line does not exist (cusp, corner, end point), or when the tangent line is vertical.. If x0 is not a critical point its derivative exists and either f ‘(x0) > 0, or f ‘(x) < 0

7. Exercise 1 The first coordinate of the critical points of each of the functions below are identified at the top of each graph. Refer to the definition of a critical point to explain why it is a critical point. Identify the type of critical point (f ’=0 or f ’ undefined)

9. Exercise http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_as_a_function.html • Identify all the critical points on the given domain • Determine the sign of the derivative between any two critical points • Estimate the derivative (draw tangent lines to find them) at x=-2, 0, 2, 4, 6 • Click on the link above and produce the graph of its derivative. • Compare your results with the applet

10. Derivative Function Given a function y=f(x) a new function is defined the following way: to each point in the domain of the function y = f(x)the value of the derivative at that point is assigned, or what is the same the value of the slope of the tangent line. This new function is called the derivative function of y = f(x). The derivative function of y=f(x) is denoted

11. Derivatives of Basic Functions • Open the down menu (bottom) and choose one of the basic functions • From the graph of the function identify • Critical points and classify them • Intervals where the derivative is positive • Intervals where the derivative is negative • Use the applet (check all the boxes) to generate the derivative function and verify your answers on part 2 • Your next task is to produce the formula for the derivative function: • Conjecture the type of graph the derivative would be • By choosing points on the graph of the derivative function produce its formula

12. More Functions To Find Their Derivative Explore the derivative function for each of the following functions and produce their formulas

13. Deriving Basic Derivative Formulas

14. If f(x)=c, constant f ‘( c )=0

15. y=m x + b , y ‘=m

16. y(x)=x2, y ‘(x)=2x

18. Derivative of a Power Function

19. Exercise 5 Rewrite each of the following functions as a power function. Use the shortcut for the derivative of power functions to find the derivative. Give the final answer with positive exponents. For each of the functions above find all their critical points

20. Basic Derivatives