Derivatives 3: The Derivative as a Function

Transcription

1 Derivatives : The Derivative as a Function 77 Derivatives : The Derivative as a Function Model : Graph of a Function g() Construct Your Understanding Questions (to do in class). Sketch lines tangent to the graph of g (above) at =, 0,, 7.,, 6,. a. Fill in the table with the slopes of these tangent lines. Choose from,., 0, 0.7,., or. Some of these values ma be used more than once, and others not at all m tangent b. Plot each of these points (below) and sketch a smooth curve connecting the points

2 78 Derivatives : Derivative as a Function c. (Check our work) The curve ou drew at the bottom of the previous page should cross the ais three times.. Look at the smooth curve ou drew in Question b at the bottom of the previous page. Consider the points on this curve between the points ou plotted, for eample, at =. a. Use this graph to estimate the value of associated with =. b. What does this value of tell ou about the original function g (in Model )?. Estimate the slope of the line tangent to g in Model at a. = 9.5 b. = Describe the method ou used to make these estimates.. According to Summar Bo D., what smbol should be used for the function ou drew on the graph at the bottom of the previous page? Eplain our reasoning. Summar Bo D.: Derivative as a Function For a function f, the function f (as defined below) gives the slope of the line tangent to the graph of f at each point where f ( ) eists. The function f is called the derivative of the function f. f ( ) lim f h f h 0 ( ) ( ) h 5. On the aes at right, sketch a portion of a function k( ) near a based on the following information about the derivative of the function: k ( a) 0 k ( ) is negative for a a k ( ) is positive for a

3 Derivatives : Derivative as a Function What can ou sa about a function f based on the information that f is a. zero at a given point? b. negative over a given interval? c. positive over a given interval? 7. Complete Summar Bo D. b filling in the blanks with increasing or decreasing, as appropriate. (Recall that increasing means that f ( ) is getting larger as gets larger.) Summar Bo D.: Positive and Negative Derivatives If f ( ) 0 on an interval then f is on that interval. If f ( ) 0 on an interval then f is on that interval. 8. On the aes at right, sketch a function g( ) based on the information that: g ( a) 0 g ( ) 0 when a (that is, the derivative is positive on both sides of a ) a

4 80 Derivatives : Derivative as a Function 9. The graph of a function g is shown at right. Mark each point where g ( ) 0. Label the intervals + if g ( ) 0 or if g ( ) 0, as appropriate. i. 0. On Graph iii, sketch the derivative of f shown on Graph i. On Graph iv, sketch the derivative of f shown on Graph ii = f() ii = f() = f '() = f '() iii. - iv. -. Circle the formula that represents the function ou drew on Graph iii. Put a bo around the formula that represents the function ou drew on Graph iv. Choose from: f ( ), f ( ), f ( ), f ( ), f ( ) f( ) 0, f( ), f( ), f ( ), f ( ), f ( ), f ( ),. (Check our work) Eplain wh the graph of the derivative of an linear function is a horizontal line.

5 Derivatives : Derivative as a Function 8 Etend Your Understanding Questions (to do in or out of class). Which graph below (vi, vii, or viii) describes f when f is the parabola shown in Graph v? Hint: start b finding the points where ou epect the derivative of the function represented in Graph v to be zero or change sign (e.g., from positive to negative). v. = f() = vi vii. - viii. -. (Check our work) The slope of the tangent line to the parabola in Graph v is a. at. On the graph of f that ou chose, what is f ( ) b. at. On the graph of f that ou chose, what is f () c. zero at what value of? On the graph of f that ou chose, what is f at that value of?

6 8 Derivatives : Derivative as a Function 5. Graph vii from the previous page (also shown below, left) is not the correct answer to Question. On the aes below, right, sketch a graph of a function f whose derivative f is represented in Graph vii. vii f () i. f() a. (Check our work) Find a point on our sketch of the graph of f where the slope of the tangent line is negative. Does this match the information on Graph vii? Eplain. b. (Check our work) Find a point on our sketch of the graph of f where the slope of the tangent line is positive. Does this match the information on Graph vii? Eplain. 6. (Check our work) The two graphs below are both correct answers to the previous question. = f () = f () i. a. Construct an eplanation for wh both answers are acceptable.

7 Derivatives : Derivative as a Function 8 b. It turns out there are an infinite number of correct answers to Question 5. Sketch several of these, and eplain wh shifting the parabola up or down does not change the graph of f (Graph vii). = f () ii. 7. Assume f is increasing over an interval. Select all that are TRUE. a. f is positive on that interval b. f is increasing on that interval c. the graph of f is below the ais on that interval d. f is negative on that interval e. the graph of f is above the ais on that interval f. f is decreasing on that interval g. f is positive on that interval 8. (Check our work) Onl two statements in the previous question are true (and statement b is not one of them). Eplain wh statement b. is false, and support our eplanation b drawing an eample of a function f that is increasing while its derivative, f, is decreasing.

8 8 Derivatives : Derivative as a Function Notes