Derivatives 2: The Derivative at a Point

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1 Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned how to find average velocit between two points and instantaneous velocit at a point. Some ke conclusions were: f ( a h) f ( a) f () t P a h Q A The slope of a line through a point P and another point Q on the graph of a position function f is equal to the average velocit over this interval. (We call this tpe of line a secant line.) Construct Your Understanding Questions (to do in class) t The slope of the line tangent to the graph of f at P is equal to the instantaneous velocit at P. The slope of this tangent line is given b a special limit called a derivative. 1. If f () t, the solid curve in Model 1, shows position versus time for a car, then a. What does the slope of the line marked A tell ou about the car? Eplain. mpq b. At which point is the car going faster P or Q? (circle one) c. The quantit h is shown as a distance along the horizontal ais. Write the label a h below the hash mark that represents the time, t a h. d. Write an epression for the slope of the secant line that goes through P and Q. (Hint: First write down the coordinates of the points P and Q.) According to Model 1, what does this slope tell ou about the movement of the car? m PQ tells ou 2. Look at the graph in Model 1 and imagine h getting smaller and smaller, causing Q to approach P. When h is infinitesimall small what does the quantit m PQ tell ou about the movement of the car? Eplain our reasoning. 3. Use our equation for m PQ to generate an epression for the quantit in Question 2.

2 70 Derivatives 2: The Derivative at a Point 4. (Check our work) Does our answer Question 3 match Summar Bo D2.1? If not, revise our answer to Question 3 or show it is equivalent to the equation below. Summar Bo D2.1: Definition of the Derivative at a Point For a function f, the derivative at a point, f ( a), can be epressed as f( a) lim f ah f a h 0 ( ) ( ) h 5. Use Summar Bo D2.1 to find f (2) for the function, f ( ) 2. Show our work. f ( ) f( a) 6. Use the formula f ( a) lim a a 2 f ( ). Show our work. (not the one in Summar Bo D2.1) to find f (2) for 7. (Check our work) The formula in Question 6 (from Activit D1) is equivalent to the formula in Summar Bo D2.1. Are our answers to Questions 5 and 6 the same? If not, go back and check our work.

3 Derivatives 2: The Derivative at a Point The graph at right shows a heating function, H() t, for a test of a potter oven. a. How man hours does it take to heat the oven from room temperature (25 o C) at t 0 to its maimum temperature of 925 o C? (Round our answer to the nearest hour.) Oven Temperature ( C) 900 H() t Time (hours) t b. What is the average temperature change per hour during oven heating? c. Is H (2) positive ( ), negative ( ), or zero (0)? What does this tell ou about the oven at t 2? d. Is H (4) positive ( ), negative ( ), or zero (0)? What does this tell ou about the oven at t 4? e. Is H (8) positive ( ), negative ( ), or zero (0)? What does this tell ou about the oven at t 8? f. Mark the point on the graph where the oven is cooling most rapidl, and eplain our reasoning. g. In general, at a point t a, what information about the oven is conveed b the derivative of the heating function at this point, H ( a)?

4 72 Derivatives 2: The Derivative at a Point 9. Complete the following statements b filling in the blanks. (Check our work) Are these completed statements consistent with our answer to part g of the previous question? On a graph of a distance () versus time (t) function, the derivative of the function at a point gives the velocit of the object at that point. Given a graph of a temperature () versus time (t) on the heating function above, the derivative of the function at a point gives the rate of temperature change with respect to time of the oven at that point. 10. Give another eample of a derivative (an instantaneous rate of change) ou might encounter in the real world. Identif the independent variable and the dependent variable, and assign reasonable units to both of these and the derivative. It ma help to sketch a graph. Be prepared to report our group s answer as part of a whole-class discussion. Challenge: Tr to find an eample that does not use time as one of our variables. Model 2: Alternate Notations Recall that the average rate of change in with respect to between 1 and 2 is equal to [the slope 2 1 of secant line marked with an S] = = 2 1 B analog to the last of these, we represent the instantaneous rate of change in with respect to d at a using the smbols: d a This is called Leibniz notation. 2 1 f() S 1 a 2 The following are equivalent was of representing f ( a), the derivative of a function f () at a (assuming the dependent variable is ): d d a a df d Construct Your Understanding Questions (to do in class) 11. On the graph in Model 2, sketch the tangent line to f( ) at a. 12. Epress the following using Leibniz notation: a. The slope of the tangent line ou drew in the previous question. b. a lim (at the point where 1 ) 0

5 Derivatives 2: The Derivative at a Point For a function f( ) that is continuous at a, circle the letter of each of the following that is equivalent to f ( a). a. The slope of the line tangent to the graph of f( ) at a b. lim f ( ) a c. lim f ( ) 0 d. lim f ( ) h0 e. The derivative of f at = a f. f ( ) f ( a) lim a a g. The instantaneous rate of change of with respect to at = a h. The instantaneous rate of change of with respect to at = a i. The instantaneous velocit at = a (where = time, and f ( ) = distance) j. The instantaneous rate of temperature change at = a degrees Celsius (where is time in hours, and ( ) f is temperature in degrees Celsius) k. f a h f a lim h 0 ( ) ( ) h l. d d a m. df d a n. ' a o. The slope of a line that passes through the point (, ( )) a f a but does not intersect the graph of the function at an other point

6 74 Derivatives 2: The Derivative at a Point Etend Your Understanding Questions (to do in or out of class) 14. (Check our work) The last statement in the previous question illustrates two common misconceptions about tangent lines. Write FALSE net to each of the two statements below. i. A tangent line can onl intersect the graph of a function once. ii. An line that intersects the graph of a function eactl once must be the tangent line to the graph at that point. a. Draw the graph of a function and a tangent line at = a such that the tangent line onl intersects the graph once. b. Draw the graph of a function and a tangent line at = a such that the tangent line intersects the graph of the function at = a, but also at one or more other points. c. Draw the graph of a function and a line that is not a tangent line to the graph of the function at = a, but intersects the graph eactl once (at = a).

7 Notes Derivatives 2: The Derivative at a Point 75

8 76 Derivatives 2: The Derivative at a Point Notes