4/16/2015 Assignment Previewer

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1 Practice Exam # 3 ( ) ( ) Due: Thu Apr :59 PM PDT Question Question Details SCalcET [ ] Use the definitions of the hyperbolic functions to find each of the following limits. (a) lim tanh x x (b) lim tanh x x (c) lim sinh x x (d) lim sinh x x (e) lim sech x x (f) lim coth x x (g) lim coth x x 0 + (h) lim coth x x 0 (i) lim csch x x 1/14

2 2. Question Details SCalcET [ ] Find the critical numbers of the function. (Enter your answers as a comma separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.) f(θ) = 18 cos θ + 9 sin 2 θ θ = 3. Question Details SCalcET [ ] Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = t 64 t 2, [ 1, 8] absolute minimum value absolute maximum value f(t) = t 64 t 2, [ 1, 8]. 1 f '(t) = t (64 t 2 ) 1/2 ( 2t) + (64 t 2 ) 1/2 t 2 1 = + = = t 2 64 t 2 t2 + (64 t2 ) 64 2t2 64 t 2 64 t 2 f '(t) = t 2 = 0 t 2 = 32 t = ± 32, but t = 32 is not in the given interval, [ 1, 8]. f '(t) does not exist if 64 t 2 = 0 is not in the given interval. f( 1) = 3 7, f( 32) = 32, and f(8) = 0. So f( 32) = 32 is the absolute maximum value and f( 1) = 3 7 is the absolute minimum value. t = ±8, but 8 4. Question Details SCalcET [ ] Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = ln x, [1, 3] Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem. Yes, f is continuous on [1, 3] and differentiable on (1, 3). No, f is not continuous on [1, 3]. No, f is continuous on [1, 3] but not differentiable on (1, 3). There is not enough information to verify if this function satisfies the Mean Value Theorem. If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a comma separated list. If it does not satisfy the hypotheses, enter DNE). c = f(x) = ln x, [1, 3]. f(b) f(a) f '(c) = b a f is continuous and differentiable on (0, ), so f is continuous on [1, 3] and differentiable on (1, 3). 1 f(3) f(1) ln 3 0 ln 3 2 = = = c =, which is in (1, 3). c ln 3 2/14

3 5. Question Details SCalcET [ ] Sketch the graph of a function that satisfies all of the given conditions. f '(0) = f '(2) = f '(4) = 0, f '(x) > 0 if x < 0 or 2 < x < 4, f '(x) < 0 if 0 < x < 2 or x > 4, f ''(x) > 0 if 1 < x < 3, f ''(x) < 0 if x < 1 or x > 3 3/14

4 f '(0) = f '(2) = f '(4) = 0 horizontal tangents at x = 0, 2, 4. f '(x) > 0 if x < 0 or 2 < x < 4 f is increasing on (, 0) and (2, 4). f '(x) < 0 if 0 < x < 2 or x > 4 f is decreasing on (0, 2) and (4, ). f ''(x) > 0 if 1 < x < 3 f is concave upward on (1, 3). f ''(x) < 0 if x < 1 or x > 3 3. f is concave downward on (, 1) and (3, ). there are inflection points where x = 1 and 4/14

5 6. Question Details SCalcET [ ] Consider the function below. f(x) = earctan 4x (a) Find the vertical asymptote(s). (Enter your answers as a comma separated list. If an answer does not exist, enter DNE.) x = Find the horizontal asymptote(s). (Enter your answers as a comma separated list. If an answer does not exist, enter DNE.) y = (b) Find the interval where the function is increasing. (Enter your answer using interval notation.) (c) Find the local maximum and minimum values. (If an answer does not exist, enter DNE.) local maximum value local minimum value (d) Find the interval where the function is concave up. (Enter your answer using interval notation.) Find the interval where the function is concave down. (Enter your answer using interval notation.) Find the inflection point. (x, y) = 7. Question Details SCalcET MI. [ ] Find the limit. Use l'hospital's Rule if appropriate. If there is a more elementary method, consider using it. ln 3x lim x 3x 5/14

6 8. Question Details SCalcET [ ] Find the limit. Use l'hospital's Rule where appropriate. If there is a more elementary method, consider using it. cos x lim x (π/2) + 1 sin x This limit has the form 0. 0 cos x lim x (π/2) + 1 sin x lim x (π/2) + sin x cos x = lim tan x =. x (π/2) + 9. Question Details SCalcET [ ] Find the limit. Use l'hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim sin x ln 3x x Question Details SCalcET MI. [ ] Find the limit. Use l'hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim (9x ln x) x 11. Question Details SCalcET [ ] Find the limit. Use l'hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim (5x + 1) cot x x /14

7 12. Question Details SCalcET [ ] Use the guidelines of this section to sketch the curve. y = x x Question Details SCalcET [ ] Use the guidelines of this section to sketch the curve. 7/14

8 y = x 2 + 4x x 8/14

9 14. Question Details SCalcET [ ] Use the guidelines of this section to sketch the curve. 3 y = x Question Details SCalcET [ ] Use the guidelines of this section to sketch the curve. y = x ln x 9/14

10 10/14

16. Question Details SCalcET7 4.7.016. [1642180] A rectangular storage container with an open top is to have a volume of 10 m 3. The length of this base is twice the width.

11 16. Question Details SCalcET [ ] A rectangular storage container with an open top is to have a volume of 10 m 3. The length of this base is twice the width. Material for the base costs $15 per square meter. Material for the sides costs $9 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.) $ Question Details SCalcET MI. [ ] A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See the figure below.) If the perimeter of the window is 24 ft, find the value of x so that the greatest possible amount of light is admitted. x = ft 11/14

12 18. Question Details SCalcET [ ] Consider the equation below. f(x) = x 7 ln x (a) Find the interval on which f is increasing. (Enter your answer using interval notation.) Find the interval on which f is decreasing. (Enter your answer using interval notation.) (b) Find the local minimum value of f. (c) Find the inflection point. (x, y) = Find the interval on which f is concave up. (Enter your answer using interval notation.) Find the interval on which f is concave down. (Enter your answer using interval notation.) 19. Question Details SCalcET [ ] Use the guidelines of this section to sketch the curve. y = x 3 12x x 12/14

13 y = f(x) = x 3 12x x = x(x 2 12x + 36) = x(x 6) 2 A. f is a polynomial, so D =. B. x intercepts are 0 and 6, y intercepts = f(0) = 0 C. No symmetry D. No asymptote E. f '(x) = 3x 2 24x + 36 = 3(x 2 8x + 12) = 3(x 2)(x 6) < 0 2 < x < 6, so f is decreasing on (2, 6) and increasing on (, 2) and (6, ). F. Local maximum value f(2) = 32, local minimum value f(6) = 0 G. f ''(x) = 6x 24 = 6(x 4) > 0 x > 4, so f is CU on (4, ) and CD on (, 4). IP at (4, 16) H. 13/14

14 20. Question Details SCalcET [ ] Find the dimensions of a rectangle with area 1,728 m 2 whose perimeter is as small as possible. (If both values are the same number, enter it into both blanks.) m (smaller value) m (larger value) Assignment Details Name (AID): Practice Exam # 3 ( ) ( ) Submissions Allowed: 100 Category: Homework Code: Locked: No Author: Mkrtchyan, Tigran ( mkrtcht@lamission.edu ) Last Saved: Apr 16, :11 PM PDT Permission: Protected Randomization: Person Which graded: Last Feedback Settings Before due date Question Score Assignment Score Publish Essay Scores Question Part Score Mark Add Practice Button Help/Hints Response Save Work After due date Question Score Assignment Score Publish Essay Scores Key Question Part Score Solution Mark Add Practice Button Help/Hints Response 14/14

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1 of 6 7/10/2013 8:32 PM 2.2Basic Differentiation Rules and Rates of Change (2047326) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1. Question Details LarCalc9 2.2.003.