JANE PROFESSOR WW Prob Lib1 Summer 2000

Transcription

1 JANE PROFESSOR WW Prob Lib Summer 2000 Sample WeBWorK problems. WeBWorK assignment Derivatives due 2//06 at 2:00 AM..( pt) If f x 9, find f 0. 2.( pt) If f x 7x 27, find f 5. 3.( pt) If f x 7 4x 5x 2, find f 5. 4.( pt) If f x 3 x 2, find f 4. 5.( pt) Let f 3 6.( pt) Let f x 3 f x 5x x 3 Use te limit definition of te derivative on page 56 to find (i) f 0 (ii) f 2 (iii) f 5 (iv) f 7 To avoid calculating four separate limits, I suggest tat you evaluate te derivative at te point wen x a. Once you ave te derivative, you can just plug in tose four values a to get te answers. 7.( pt) For eac of te given functions f x, find te derivative f c at te given point c, using Teorem, first finding a f c. f x 4x 9x 9 ;c 3 a = f c = f x x 2 8x 22 on te interval 4 ;c 7 a = f c = 8.( pt) Te position of a cat running from a dog down a dark alley is given by te values of te table. t(seconds) s(feet) A. Find te average velocity tor te time period beginning wen t=2 and lasting. 3 s 2. 2 s 3. s B. Draw te grap of te function yourself and estimate te instantaneous velocity wen t=2 9.( pt) Tis problem tests calculating new functions from old ones: From te table below calculate te quantities asked : x f x g x f x g x f 2 g 2 5 f f 3 f g 2 If x g f x, calculate 3. 0.( pt) Constructing new functions from old ones and calculating te derivative of te new function from te derivatives of te old functions: From te table below calculate te quantities asked : x f x g x f x g x

2 f g 4 f g 4 f g 4 f 4 g 4 5.( pt) Tis problem tests calculating new functions from old ones: From te table below calculate te quantities asked : x f x g x f x g x f g 3 If x f g x, calculate 3 f f 3 f g If x f f x, calculate Tis problem tests calculating new functions from old ones: From te table below calculate te quantities asked : Given te following table: 2.( pt) x f x g x f x g x ( pt) x f(x) f g 4 If x g f x, calculate 3. If x f g x, calculate 3 f f 3 f g 3 Calculate te value of f = to two places of accuracy. To obtain more precise inmation about te value of f near enter a new increment value x ere rulein.0in and ten press te Submit Answer button. How will you tell wen your increment is small enoug to give you a good answer te problem? 2

3 4.( pt) Identify te graps A (blue), B( red) and C (green) as te graps of a function and its derivatives: is te grap of te function is te grap of te function s first derivative is te grap of te function s second derivative 5.( pt) Identify te graps A (blue), B( red) and C (green) as te graps of a function and its derivatives: is te grap of te function is te grap of te function s first derivative is te grap of te function s second derivative 3 6.( pt) Identify te graps A (blue), B( red) and C (green) as te graps of a function and its derivatives: is te grap of te function is te grap of te function s first derivative is te grap of te function s second derivative 7.( pt) Let f x 2x 3 9x 5 Use te limit definition of te derivative on page 63 to calculate te derivative of f : f x. Use te same mula from above to calculate te derivative of tis new function (i.e. te second derivative of f ): f x. 8.( pt) Te oracle function f x is presented below. For eac x value you enter te oracle will tell you te value f x. Calculate te derivative of te function at 5 using te Newton quotient definition. f x at 5 = You can use a calculator x f(x) Remember te tecnique finding instantaneous velocities from average velocities? Tis is te same ting. 9.( pt) Below is an oracle function. An oracle function is a function presented interactively. Wen you type in a t value, and press te f button te value f t appears in te rigt and window. Tere are tree lines, so you can calculate tree different values of te function at one time. Te function f(t) represents te eigt in feet of a ball trown into te air, t seconds after it as been trown. Calculate te average velocity 3.08 seconds after te ball as been trown. Average velocity at 3 08 = You can use a calculator

4 Te java Script calculator was displayed ere Remember tis tecnique finding velocities. Later we will use te same metod to find te derivative of functions suc as f t. a b 20.( pt) Find a and b suc tat te function f x is differentiable everywere. x 2 6x 3 if x 2 ax b if x 2 2.( pt) Let f x be te function x 2 5x 8. Ten te quotient f 3! " f 3" a can be simplified to a b : and b 22.( pt) Let f x be te function x! 7. Ten te quotient f 5! " f 5" can be simplified to a! b : a and b 23.( pt) Let f x x 3 x. Calculate te difference quotient f 2! " f 2" # # 0 # $ 0 # $ If someone now told you tat te derivative (slope of te tangent line to te grap) of f x at x 2 was an integer, wat would you expect it to be? 24.( pt) Let f x x 6. Calculate te difference quotient f 4! " f 4" # # 0 # $ 0 # $ If someone now told you tat te derivative (slope of te tangent line to te grap) of f x at x 4 was n 2 some integer n wat would you expect n to be? n 25.( pt) Let f x x 5. Calculate te difference quotient f! " f " # # 0 # $ 0 # $ 4 If someone now told you tat te derivative (slope of te tangent line to te grap) of f x at x was n some integer n wat would you expect n to be? n 26.( pt) Graps A and B are approximate graps of f and f f x x 2 8x 9. So f is decreasing (and f is negative) on te interval a a.

5 27.( pt) 28.( pt) Graps A and B are approximate graps of f and f f x x 2 0x 26. So f is increasing (and f is positive) on te interval a a. Graps A and B are approximate graps of f and f f x x 2 x 9. So f is decreasing (and f is negative) on te interval 0 a a. % Prepared by te WeBWorK group, Dept. of Matematics, University of Rocester, c UR 5