This file is /conf/snippets/setheader.pg you can use it as a model for creating files which introduce each problem set.

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1 Victoria Howle WeBWorK assignment number WW06 is due : 03/30/2012 at 08:07pm CDT. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information. spr12vhowlem1451s008 This file is /conf/snippets/setheader.pg you can use it as a model for creating files which introduce each problem set. The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help. Don t spend a lot of time guessing it s not very efficient or effective. Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2 3 instead of 8, sin(3 pi/2)instead of -1, e (ln(2)) instead of 2, (2 +tan(3)) (4 sin(5)) 6 7/8 instead of , etc. Here s the list of the functions which WeBWorK understands. You can use the Feedback button on each problem page to send to the professors. 1. (1 pt) Library/270/setDerivatives8RelatedRates/s2 8 7.pg A street light is at the top of a 11 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 ft from the base of the pole? (1 pt) Library/270/setDerivatives8RelatedRates/s2 8 21a.pg Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 13 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V = 1 3 πr2 h Note: See number 21 on pg 258 for a picture of this (1 pt) Library/270/setDerivatives8RelatedRates/s2 8 5.pg A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.3 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 11 cm. (Note the answer is a positive number) (1 pt) Library/270/setDerivatives8RelatedRates/s2 8 3.pg Let and let Find dx dt when x = xy = 2 dy dt = 5 5. (1 pt) 1351lib/sbs ch3/sec 3 7/sbs-s37-4.pg This is problem 4 Section 3.7 page 162. If 4x 2 y = 100 and dy/dt = 6, find dx/dt when x = 1. dx/dt = (1 pt) 1351lib/sbs ch3/sec 3 7/sbs-s37-14.pg This is problem 14 Section 3.7 page 162. A rock is dropped into a lake and an expanding circular ripple results. When the radius of the ripple is 8 in., the radius is increasing at a rate of 3 in./sec. At what rate is the area enclosed by the ripple changing at this time? da/dt = sq. in./sec

2 7. (1 pt) Library/270/setDerivatives9Approximations/s pg Use linear approximation, i.e. the tangent line, to approximate as follows: Let f (x) = x 6. The equation of the tangent line to f (x) at x = 3 can be written in the form y = mx + b where m is: and where b is: Using this, we find our approximation for is (1 pt) Library/270/setDerivatives9Approximations/c2s9p6.pg The circumference of a sphere was measured to be cm with a possible error of cm. Use linear approximation to estimate the maximum error in the calculated surface area. Estimate the relative error in the calculated surface area (1 pt) Library/270/setDerivatives9Approximations/s pg Use linear approximation, i.e. the tangent line, to approximate 36.4 as follows: Let f (x) = x. The equation of the tangent line to f (x) at x = 36 can be written in the form y = mx + b where m is: and where b is: Using this, we find our approximation for 36.4 is NOTE: For this part, give your answer to at least 9 significant figures or use fractions to give the exact answer (1 pt) Library/270/setDerivatives9Approximations/s pg Let y = 2x 2. Find the change in y, y when x = 1 and x = 0.1 Find the differential dy when x = 1 and dx = (1 pt) Library/270/setDerivatives9Approximations/s2 9 7.pg Let y = 5x 2 + 8x + 4. Find the differential dy when x = 1 and dx = 0.4 Find the differential dy when x = 1 and dx = (1 pt) 1351lib/sbs ch3/sec 3 8/sbs-s38-2.pg This is problem 2 Section 3.8 page 173. Find the differential d(3 5x 2 ) = dx. -10*x 13. (1 pt) 1351lib/sbs ch3/sec 3 8/sbs-s38-6.pg This is problem 6 Section 3.8 page 173. Find the differential d(x sin(2x)) = dx. 2*x*cos(2*x)+sin(2*x) 14. (1 pt) 1351lib/sbs ch4/sec 4 1/sbs-s41-8.pg This is problem 8 Section 4.1 page 193. Find the critical values for g(t) = 3t 5 20t 3 on the interval [ 1, 2]. Then decide whether the critical value is a maximum (MAX), minimum (MIN) or neither (N). For this problem there are three critical values t 1 < t 2 < t 3 and endpoints in the given interval: t 1 = ( MAX, MIN, N ) t 2 = ( MAX, MIN, N ) t 3 = ( MAX, MIN, N ) -1 MAX 0 N 2 MIN 15. (1 pt) 1351lib/sbs ch4/sec 4 1/sbs-s41-12.pg This is problem 12 Section 4.1 page 193. Find the critical values for f (x) = x 3 on the interval [ 4, 4]. Then decide whether the critical value is a maximum

3 (MAX), minimum (MIN) or neither (N). For this problem there are three critical values x 1 < x 2 < x 3 and endpoints in the given interval: x 1 = ( MAX, MIN, N ) x 2 = ( MAX, MIN, N ) x 3 = ( MAX, MIN, N ) -4 MAX 3 MIN 4 N 16. (1 pt) 1351lib/sbs ch4/sec 4 1/sbs-s41-20.pg This is problem 20 Section 4.1 page (1 pt) 1351lib/sbs ch4/sec 4 2/sbs-s42-6.pg This is problem 6 Section 4.2 page 199. Verify that the function f satisfies the hypotheses of the MVT on the given interval [a,b]. Then find all numbers c between a and b for which f (b) f (a) b a = f (c). f (x) = 2x 3 x 2, on [0,2] Find the absolute maximum and absolute minimum of g(x) = 2x 3 3x 2 36x + 4 on the interval [ 4,4]. To do this problem you will be asked to carry out some intermediate steps: (1) Find f (x) = Absolute Maximum: x = g(x) = Absolute Minimum: x = g(x) = (2) Find f (b) f (a) b a = (1 pt) 1351lib/sbs ch4/sec 4 1/sbs-s41-25.pg This is problem 25 Section 4.1 page 194. Find the absolute maximum and absolute minimum of s(t) = t cos(t) sin(t) on the interval [0,2π]. (3) There is one value of c : c = 6*x**2-2*x (1 pt) 1351lib/sbs ch4/sec 4 3/sbs-s43-12.pg This is problem 12 Section 4.3 page 215. Absolute Maximum: t = s(t) = Absolute Minimum: t = s(t) = Given f (x) = 1 3 x3 9x + 2 (a) Find all critical numbers. If there are no such real x, type DNE in the answer blank. If there is more than one real x, give

4 a comma separated list (e.g. 1,2). MAKE SURE TO ENTER THE NUMBERS IN INCREASING ORDER. ANSWER (critical points) : ANSWER (Increasing) : ANSWER (Decreasing) : (b) Find where the function is increasing and decreasing. ANSWER (Increasing) : ANSWER (Decreasing) : (c) Find where the function is concave up and concave down. ANSWER (concave up): ANSWER (concave down): (d) Find the x coordinate all points of inflection. If there are no such real x, type DNE in the answer blank. If there is more that one real x, give a comma separated list (e.g. 1,2). MAKE SURE TO ENTER THE NUMBERS IN INCREASING OR- DER. ANSWER (inflection points): -3, 3 (-infinity,-3) U (3,infinity) (-3,3) (0,infinity) (-infinity,0) (1 pt) 1351lib/sbs ch4/sec 4 3/sbs-s43-21.pg This is problem 21 Section 4.3 page 215. (b) Determine where the function is concave up and concave down. ANSWER (concave up): ANSWER (concave down): (-infinity,-3) U (3,infinity) (-3,0) U (0,3) (0,infinity) (-infinity,0) 21. (1 pt) 1351lib/sbs ch4/sec 4 3/sbs-s43-32.pg This is problem 32 Section 4.3 page 215. Given t(θ) = sin(θ) 2 cos(θ) on the interval [0, 2π]. (a) Determine where the function is increasing and decreasing. ANSWER (Increasing) : ANSWER (Decreasing) : Given f (x) = 1 + 2x + 18/x (a) Determine where the function is increasing and decreasing. 4 (b) Determine where the function is concave up and concave down. If the function is either not concave up (or concave

5 down) then simply enter NONE -1.5 ANSWER (concave up): ANSWER (concave down): [0, ) U ( , ] ( , ) [0, ) U ( , ] ( , ) 22. (1 pt) Library/270/setDerivatives11Newton/s a.pg Use Newton s method to approximate a root of the equation cos(x 2 + 5) = x 3 as follows. Let x 1 = 1 be the initial approximation. The second approximation x 2 is The third approximation x 3 is (1 pt) Library/270/setDerivatives11Newton/s pg Use Newton s method to approximate a root of the equation x 3 + x + 3 = 0 as follows. Let x 1 = 1 be the initial approximation. The second approximation x 2 is and the third approximation x 3 is (1 pt) Library/UVA-Stew5e/setUVA-Stew5e-C04S09- NewtonsMethod/ pg Use Newton s method to approximate the value of (1 pt) Library/Utah/AP Calculus I/set4 Graphing and Maximum- Minimum Problems/1210s7p9.pg Consider the function f (x) = 4x 3 3x on the interval [ 3,3]. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there exists at least one c in the open interval ( 3,3) such that f (c) is equal to this mean slope. For this problem, there are two values of c that work. smaller one is and the larger one is (1 pt) Library/ma122DB/set7/s4 2 5.pg Consider the function f (x) = 7 7x 2/3 on the interval [ 1,1]. The Which of the three hypotheses of Rolle s Theorem fails for this function on the inverval? (a) f (x) is continuous on [ 1,1]. (b) f (x) is differentiable on ( 1,1). (c) f ( 1) = f (1). as follows: Let x 1 = 2 be the initial approximation. The second approximation x 2 is and the third approximation x 3 is (1 pt) Library/270/setDerivatives12MVT/s pg Consider the function f (x) = 7 8x 2 on the interval [ 6,3]. Find the average or mean slope of the function on this interval, i.e. f (3) f ( 6) = 3 ( 6) By the Mean Value Theorem, we know there exists a c in the open interval ( 6,3) such that f (c) is equal to this mean slope. For this problem, there is only one c that works. Find it. Answer:(a, b, or c ) b 28. (1 pt) Library/270/setDerivatives10MaxMin/s pg Consider the function f (x) = 1 2x 2, 5 x 1. The absolute maximum value is and this occurs at x equals The absolute minimum value is and this occurs at x equals

6 29. (1 pt) Library/270/setDerivatives1/nsc2s10p2.pg Identify the graphs A (blue), B( red) and C (green) as the graphs of a function and its derivatives: is the graph of the function is the graph of the function s first derivative is the graph of the function s second derivative B C A Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 6