Main Points: 1. Limit of Difference Quotients. Prep 2.7: Derivatives and Rates of Change. Names of collaborators:
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1 Name: Section: Names of collaborators: Main Points:. Definition of derivative as limit of difference quotients. Interpretation of derivative as slope of grap. Interpretation of derivative as instantaneous rate of cange. Limit of Difference Quotients Recall from Prep., tat to estimate te slope of a tangent line, we use te slopes of secant lines. In particular, to estimate te slope of te tangent line to te grap of a function f(x) at a point x = a in te domain of f, te slope of te secant line troug P = (a, f(a)) and Q = (a +, f(a + )) is: D = y x = f(a + ) f(a) Tis is called te difference quotient. Te value of D depends on. Te limit of D() as approaces zero (if te limit exists) is te slope of te tangent line. In te preparatory assignment on limits, you explored tis idea numerically, by creating tables of D-values for smaller and smaller values of. Definition For a function f(x) and a number a in its domain, te derivative of f at a, denoted f (a), is: provided tat te limit exists. Exercise f (a) = lim 0 D() = lim 0. Look back at Prep. and Prep. to answer tese questions. f(a + ) f(a) (a) If f(x) = x and a =, wat is a good estimate for f (a)? (b) If f(x) = x and a =, wat can you say about f (a)?
2 . Find an equation of te tangent line to te grap of y = f(x) at x = if f() = and f () =.. Suppose tat y = x is an equation of te tangent line to te curve y = f(x) at te point x =. Find f() and f (). Having discussed ow to evaluate limits symbolically (as in.), we can determine te exact value for f (a), at least in some relatively simple examples. (See Example on page 7.) Exercise. Let f(x) = x and a =. Find te exact value of f (a) by evaluating te following limit symbolically as in Section.: + lim 0 (You will need to rationalize te numerator by multiplying numerator and denominator by te conjugate + +. See Section., Example 6, page 0.)
3 . Interpretation as Rates of Cange Recall tat, in applications, we interpret slopes and difference quotients as rates of cange. In particular, te difference quotient is an average rate of cange. Te limit of te difference quotient, te derivative, is te instantaneous rate of cange. In summary, we can interpret te difference quotient and te derivative in tese ways: difference quotient slope of secant average velocity average rate of cange derivative slope of tangent instantaneous velocity instantaneous rate of cange For examples of interpretating difference quotients and derivatives, see Examples 6 and 7, p 8-9. Exercises. Look back at Exercise from Prep.. Estimate te instantaneous velocity of te buoy at t = 0. (Make sure to include units.) 6. Let T (t) be te temperature (in F) in Poenix t ours after midnigt on September 0, 008. Te table sows values of tis function recorded every two ours. t T (a) Look back at your Exercise from Prep., and give an estimate T (8). (b) Wat is te meaning of T (8)? Use a complete sentence, and make sure to include units. (c) Use (b) to estimate te temperature at 9:00 am.
4 7. Te number of bacteria after t ours in a controlled laboratory experiment is n = f(t). (a) Wat is te meaning of te statement f () = 00? Wat are te units of f ()? (b) Suppose tere is an unlimited amount of space and nutrients for te bacteria. Wic do you tink is larger, f () or f (0)? Wy? 8. Sown are graps of te position functions of two runners, wo run a 00 meter race. 00 Position (meters) from te starting line 7 0 Frances Mary Time (seconds) since start of race (a) Describe and compare ow te runners run te race. (In particular, wo won te race?) (b) At wat time is te distance between te runners te greatest? (c) At wat time do tey ave te same velocity?
5 9. A duck starts by waddling nort on E; te grap of its position function is sown. Position (meters) from starting point 0 6 Time (seconds) since start (a) Wen is te duck waddling nort? (b) Waddling sout? (c) Standing still? (d) Draw a grap of te velocity function on te axes below. Velocity (meters per second) Time (seconds) since start
For a function f(x) and a number a in its domain, the derivative of f at a, denoted f (a), is: D(h) = lim
Name: Section: Names of collaborators: Main Points: 1. Definition of derivative as limit of difference quotients 2. Interpretation of derivative as slope of graph 3. Interpretation of derivative as instantaneous
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