Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.

Transcription

1 Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite justifiction when necessry. lim x 2 x + 2 x b e x e x lim x 0 sin(x) c lim x 0 x tn(5x) d lim x x 2 1 8x 2 + 8x

2 Problem 2 Compute dy dx y = x x 2 csc 1 (x) for the following functions: b y = (2x + 5)3 e x + 7 c y = sin 1 (x 2 cos(x)) d y = (x 2 + 1) 4x e y = ln(x) tn(x)

3 Problem 3 Let f(x) = 5x + 3 Use the definition of the derivtive to compute f (x). b Find the eqution of the tngent line to f(x) t the point x = 2.

4 Problem 4 Find the verge rte of chnge of the function f(x) = 3x + 1 between x = 1 nd x = 5 b Find the point c in the intervl [1, 5] where f (c) equls the verge rte computed in prt.

5 Problem 5 In ech prt of this problem, x nd y stisfy x 2 xy + y 2 = 9. Find dy dx when (x, y) = (0, 3). b Find ll points where the tngent line is horizontl.

6 Problem 6 Stte the Men Vlue Theorem. b Given tht function f(x) stisfies the hypotheses of the Men Vlue Theorem for the intervl [ 3, 2], nd we hve dt f(2) = 11 nd f (x) <= 4 for ll x [ 3, 2], wht is the smllest possible vlue of f( 3)?

7 Problem 7 A certin function stisfies f(7) = 3 f (7) = 5 f (7) = 10 Use liner pproximtion to estimte the vlue f(6.9). b Would you expect your nswer to overestimte or underestimte the true vlue? Explin.

8 Problem 8 sec(7x + 2) tn(7x + 2)dx Work out the following ntiderivtives using ny technique tht you like. b x (x 2 dx + 4) 17 c 7x 2dx d x x dx

9 Problem 9 Consider trpezoid sketched in the coordinte plne. Two vertices of the trpezoid re locted t ( 2, 0) nd (2, 0), nd the other two lie on the semicircle given by y = 4 x 2. Wht is the mximum possible re of such trpezoid? (Note: the re of ny trpezoid with bses b 1 nd b 2 nd height h is (1/2)h(b 1 + b 2 )).

10 Problem 10 A 20 foot rope is tied to the bck of bot. It is then stretched over the edge of 5 foot high dock nd tied to the nkle of n elephnt who hppens to be stnding on the dock (see figure below). If the bot moves wy from the dock t 10 ft/sec, how fst is the elephnt drgged when the bot is 12 feet from the bse of the dock?

11 Problem 11 Suppose we know the following fcts bout n unknown function f(x) with domin ll rel numbers: lim = x lim = x f(0) = 7 f (0) = 0 f(2) = 1 f (2) = 0 f (x) > 0 for x < 0, x > 2 f (x) < 0 for 0 < x < 2 f (x) < 0 for x < 1 f (x) > 0 for x > 1 Find: The intervl(s) where f is incresing. b The x-coordinte(s) where f hs locl mximum. c The x-coordinte(s) where f hs locl minimum. d The intervl(s) where f is concve up. e Now grph the function below.

12 Problem 12 You re building the frme for Christms tree. The frme of the tree consists of circulr floor nd verticl centrl pole extending from the floor to the pex of the cone. The conicl cvity will be filled with 9 cubic yrds of flme-resistnt stuffing. The floor costs $2 per squre yrd, nd the pole costs $6 per yrd. Wht re the dimensions of the lest expensive frme possible? (Remember, V = (1/3)πr 2 h for cone).

13 Problem 13 Prt I Stte both prts of the Fundmentl Theorem of Clculus: b Prt II

14 Problem 14 Use prt one of the Fundmentl Theorem of Clculus to compute: d dx x 1 t2 + sin tdt b Use prt two of the Fundmentl Theorem of Clculus to compute: π/4 π/6 csc 2 (t)dt

15 Problem 15 Evlute x dx. b Drw grph of y = 1/x below, nd on tht grph, sketch in four right-endpoint rectngles to pproximte the vlue you computed in prt bove. c Now compute your re estimte using the four right-endpoint rectngles.

16 Problem 16 Find the re of the region enclosed by the two curves y = 2x, y = x 3 Note tht this region hs pieces lying in both the first nd third qudrnts.

17 Problem 17 Dr. Teddy is wering bungee cord nd rocket pck. He jumps off bridge t time t = 0. We mesure Dr. Teddy s position function in feet ABOVE the bridge, t minutes fter jumping. Dr. Teddy s ccelertion upwrd fter t minutes is given by the function: (t) = 6 sin(3t) ft/min 2 If the initil velocity is v(0) = 1 ft/min, find Dr. Teddy s velocity function v(t) in ft/min. b Given tht s(0) = 0, find Dr. Teddy s position function s(t) in ft. c When is the first time tht Dr. Teddy chnges direction?

18 Problem 18 The grph in the figure below shows the rte of chnge of the depth of smll pond in cm/dy tken over period of 9 dys. The intervl from 0 to 1 represents Dy 1, the intervl from 1 to 2 represents Dy 2, nd so on. At the beginning of Dy 1, the pond strts with depth 200 cm. Wht is the longest run of dys during which the level of the pond decresed every dy? b At the end of which dy is the depth of the pond gretest? c Is the depth of the pond greter thn or less thn 200 cm t the end of the nine dy period?