Calculus
5th Edition
ISBN: 9781429241861
Author: Laura Taalman, Peter Kohn
Publisher: W. H. Freeman
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Chapter 2.5, Problem 13C
To determine
To find: The explanation for that the logarithmic functions and the means of that
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Evaluate the limit, and show your answer to 4 decimals if necessary.
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16x18 - 16y18
429-4y⁹
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lim
xy+y³
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= 0:
Along the path y
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Along the path y = 2x:
Chapter 2 Solutions
Calculus
Ch. 2.1 - Prob. 1TBCh. 2.1 - Prob. 2TBCh. 2.1 - Prob. 3TBCh. 2.1 - Prob. 4TBCh. 2.1 - Prob. 0CCh. 2.1 - Prob. 1CCh. 2.1 - Prob. 2CCh. 2.1 - Prob. 3CCh. 2.1 - Prob. 4CCh. 2.1 - Prob. 5C
Ch. 2.1 - Prob. 6CCh. 2.1 - Prob. 7CCh. 2.1 - Prob. 8CCh. 2.1 - Prob. 9CCh. 2.1 - Prob. 10CCh. 2.1 - Prob. 11CCh. 2.1 - Prob. 12CCh. 2.1 - Prob. 13CCh. 2.1 - Prob. 14CCh. 2.1 - Prob. 15CCh. 2.1 - Prob. 16CCh. 2.1 - Prob. 17CCh. 2.1 - Prob. 18CCh. 2.1 - Prob. 19CCh. 2.1 - Prob. 20CCh. 2.1 - Prob. 21SCh. 2.1 - Prob. 22SCh. 2.1 - Prob. 23SCh. 2.1 - Prob. 24SCh. 2.1 - Prob. 25SCh. 2.1 - Prob. 26SCh. 2.1 - Prob. 27SCh. 2.1 - Prob. 28SCh. 2.1 - Prob. 29SCh. 2.1 - Prob. 30SCh. 2.1 - Prob. 31SCh. 2.1 - Prob. 32SCh. 2.1 - Prob. 33SCh. 2.1 - Prob. 34SCh. 2.1 - Prob. 35SCh. 2.1 - Prob. 36SCh. 2.1 - Prob. 37SCh. 2.1 - Prob. 38SCh. 2.1 - Prob. 39SCh. 2.1 - Prob. 40SCh. 2.1 - Prob. 41SCh. 2.1 - Prob. 42SCh. 2.1 - Prob. 43SCh. 2.1 - Prob. 44SCh. 2.1 - Prob. 45ACh. 2.1 - Prob. 46ACh. 2.1 - Prob. 47ACh. 2.1 - Prob. 48ACh. 2.1 - Prob. 49ACh. 2.1 - Prob. 50ACh. 2.1 - Prob. 51ACh. 2.1 - Prob. 52ACh. 2.1 - Prob. 53ACh. 2.1 - Prob. 54ACh. 2.1 - Prob. 55ACh. 2.1 - Prob. 56ACh. 2.1 - Prob. 57ACh. 2.1 - Prob. 58PCh. 2.1 - Prob. 59PCh. 2.1 - Prob. 1TFCh. 2.1 - Prob. 2TFCh. 2.1 - Prob. 3TFCh. 2.2 - Prob. 1TBCh. 2.2 - Prob. 2TBCh. 2.2 - Prob. 0CCh. 2.2 - Prob. 1CCh. 2.2 - Prob. 2CCh. 2.2 - Prob. 3CCh. 2.2 - Prob. 4CCh. 2.2 - Prob. 5CCh. 2.2 - Prob. 6CCh. 2.2 - Prob. 7CCh. 2.2 - Prob. 8CCh. 2.2 - Prob. 9CCh. 2.2 - Prob. 10CCh. 2.2 - Prob. 11CCh. 2.2 - Prob. 12CCh. 2.2 - Prob. 13CCh. 2.2 - Prob. 14CCh. 2.2 - Prob. 15CCh. 2.2 - Prob. 16CCh. 2.2 - Prob. 17CCh. 2.2 - Prob. 18CCh. 2.2 - Prob. 19CCh. 2.2 - Prob. 20CCh. 2.2 - Prob. 21CCh. 2.2 - Prob. 22CCh. 2.2 - Prob. 23SCh. 2.2 - Prob. 24SCh. 2.2 - Prob. 25SCh. 2.2 - Prob. 26SCh. 2.2 - Prob. 27SCh. 2.2 - Prob. 28SCh. 2.2 - Prob. 29SCh. 2.2 - Prob. 30SCh. 2.2 - Prob. 31SCh. 2.2 - Prob. 32SCh. 2.2 - Prob. 33SCh. 2.2 - Prob. 34SCh. 2.2 - Prob. 35SCh. 2.2 - Prob. 36SCh. 2.2 - Prob. 37SCh. 2.2 - Prob. 38SCh. 2.2 - Prob. 39SCh. 2.2 - Prob. 40SCh. 2.2 - Prob. 41SCh. 2.2 - Prob. 42SCh. 2.2 - Prob. 43SCh. 2.2 - Prob. 44SCh. 2.2 - Prob. 45SCh. 2.2 - Prob. 46SCh. 2.2 - Prob. 47SCh. 2.2 - Prob. 48SCh. 2.2 - Prob. 49SCh. 2.2 - Prob. 50SCh. 2.2 - Prob. 51SCh. 2.2 - Prob. 52SCh. 2.2 - Prob. 53SCh. 2.2 - Prob. 54SCh. 2.2 - Prob. 55SCh. 2.2 - Prob. 56SCh. 2.2 - Prob. 57SCh. 2.2 - Prob. 58SCh. 2.2 - Prob. 59SCh. 2.2 - Prob. 60SCh. 2.2 - Prob. 61SCh. 2.2 - Prob. 62SCh. 2.2 - Prob. 63SCh. 2.2 - Prob. 64SCh. 2.2 - Prob. 65SCh. 2.2 - Prob. 66SCh. 2.2 - Prob. 67SCh. 2.2 - Prob. 68SCh. 2.2 - Prob. 69SCh. 2.2 - Prob. 70SCh. 2.2 - Prob. 71SCh. 2.2 - Prob. 72SCh. 2.2 - Prob. 73SCh. 2.2 - Prob. 74SCh. 2.2 - Prob. 75SCh. 2.2 - Prob. 76SCh. 2.2 - Prob. 77SCh. 2.2 - Prob. 78SCh. 2.2 - Prob. 79SCh. 2.2 - Prob. 80SCh. 2.2 - Prob. 81SCh. 2.2 - Prob. 82SCh. 2.2 - Prob. 83SCh. 2.2 - Prob. 84SCh. 2.2 - Prob. 85SCh. 2.2 - Prob. 86SCh. 2.2 - Prob. 87ACh. 2.2 - Prob. 88ACh. 2.2 - Prob. 89ACh. 2.2 - Prob. 90ACh. 2.2 - Prob. 91ACh. 2.2 - Prob. 92PCh. 2.2 - Prob. 93PCh. 2.2 - Prob. 94PCh. 2.2 - Prob. 95PCh. 2.2 - Prob. 96PCh. 2.2 - Prob. 97PCh. 2.2 - Prob. 98PCh. 2.2 - Prob. 1TFCh. 2.2 - Prob. 2TFCh. 2.2 - 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Prob. 48SCh. 2.3 - Prob. 49SCh. 2.3 - Prob. 50SCh. 2.3 - Prob. 51SCh. 2.3 - Prob. 52SCh. 2.3 - Prob. 53SCh. 2.3 - Prob. 54SCh. 2.3 - Prob. 55SCh. 2.3 - Prob. 56SCh. 2.3 - Prob. 57SCh. 2.3 - Prob. 58SCh. 2.3 - Prob. 59SCh. 2.3 - Prob. 60SCh. 2.3 - Prob. 61SCh. 2.3 - Prob. 62SCh. 2.3 - Prob. 63SCh. 2.3 - Prob. 64SCh. 2.3 - Prob. 65SCh. 2.3 - Prob. 66SCh. 2.3 - Prob. 67SCh. 2.3 - Prob. 68SCh. 2.3 - Prob. 69SCh. 2.3 - Prob. 70SCh. 2.3 - Prob. 71SCh. 2.3 - Prob. 72SCh. 2.3 - Prob. 73SCh. 2.3 - Prob. 74SCh. 2.3 - Prob. 75SCh. 2.3 - Prob. 76SCh. 2.3 - Prob. 77SCh. 2.3 - Prob. 78SCh. 2.3 - Prob. 79ACh. 2.3 - Prob. 80ACh. 2.3 - Prob. 81ACh. 2.3 - Prob. 82ACh. 2.3 - Prob. 83ACh. 2.3 - Prob. 84PCh. 2.3 - Prob. 85PCh. 2.3 - Prob. 86PCh. 2.3 - Prob. 87PCh. 2.3 - Prob. 88PCh. 2.3 - Prob. 89PCh. 2.3 - Prob. 90PCh. 2.3 - Prob. 91PCh. 2.3 - Prob. 92PCh. 2.3 - Prob. 1TFCh. 2.3 - Prob. 2TFCh. 2.3 - Prob. 3TFCh. 2.3 - Prob. 4TFCh. 2.3 - Prob. 5TFCh. 2.4 - Prob. 1TBCh. 2.4 - Prob. 2TBCh. 2.4 - Prob. 0CCh. 2.4 - Prob. 1CCh. 2.4 - Prob. 2CCh. 2.4 - Prob. 3CCh. 2.4 - Prob. 4CCh. 2.4 - Prob. 5CCh. 2.4 - Prob. 6CCh. 2.4 - Prob. 7CCh. 2.4 - Prob. 8CCh. 2.4 - Prob. 9CCh. 2.4 - Prob. 10CCh. 2.4 - Prob. 11CCh. 2.4 - Prob. 12CCh. 2.4 - Prob. 13CCh. 2.4 - Prob. 14CCh. 2.4 - Prob. 15CCh. 2.4 - Prob. 16CCh. 2.4 - Prob. 17CCh. 2.4 - Prob. 18CCh. 2.4 - Prob. 19CCh. 2.4 - Prob. 20CCh. 2.4 - Prob. 21SCh. 2.4 - Prob. 22SCh. 2.4 - Prob. 23SCh. 2.4 - Prob. 24SCh. 2.4 - Prob. 25SCh. 2.4 - Prob. 26SCh. 2.4 - Prob. 27SCh. 2.4 - Prob. 28SCh. 2.4 - Prob. 29SCh. 2.4 - Prob. 30SCh. 2.4 - Prob. 31SCh. 2.4 - Prob. 32SCh. 2.4 - Prob. 33SCh. 2.4 - Prob. 34SCh. 2.4 - Prob. 35SCh. 2.4 - Prob. 36SCh. 2.4 - Prob. 37SCh. 2.4 - Prob. 38SCh. 2.4 - Prob. 39SCh. 2.4 - Prob. 40SCh. 2.4 - Prob. 41SCh. 2.4 - Prob. 42SCh. 2.4 - Prob. 43SCh. 2.4 - Prob. 44SCh. 2.4 - Prob. 45SCh. 2.4 - Prob. 46SCh. 2.4 - Prob. 47SCh. 2.4 - Prob. 48SCh. 2.4 - Prob. 49SCh. 2.4 - Prob. 50SCh. 2.4 - Prob. 51SCh. 2.4 - Prob. 52SCh. 2.4 - Prob. 53SCh. 2.4 - Prob. 54SCh. 2.4 - Prob. 55SCh. 2.4 - Prob. 56SCh. 2.4 - Prob. 57SCh. 2.4 - Prob. 58SCh. 2.4 - Prob. 59SCh. 2.4 - Prob. 60SCh. 2.4 - Prob. 61SCh. 2.4 - Prob. 62SCh. 2.4 - Prob. 63SCh. 2.4 - Prob. 64SCh. 2.4 - Prob. 65SCh. 2.4 - Prob. 66SCh. 2.4 - Prob. 67SCh. 2.4 - Prob. 68SCh. 2.4 - Prob. 69SCh. 2.4 - Prob. 70SCh. 2.4 - Prob. 71SCh. 2.4 - Prob. 72SCh. 2.4 - Prob. 73SCh. 2.4 - Prob. 74SCh. 2.4 - Prob. 75SCh. 2.4 - Prob. 76SCh. 2.4 - Prob. 77SCh. 2.4 - Prob. 78SCh. 2.4 - Prob. 79SCh. 2.4 - Prob. 80SCh. 2.4 - Prob. 81SCh. 2.4 - Prob. 82SCh. 2.4 - Prob. 83SCh. 2.4 - Prob. 84SCh. 2.4 - Prob. 85ACh. 2.4 - Prob. 86ACh. 2.4 - Prob. 87ACh. 2.4 - Prob. 88PCh. 2.4 - Prob. 89PCh. 2.4 - Prob. 90PCh. 2.4 - Prob. 91PCh. 2.4 - Prob. 92PCh. 2.4 - Prob. 93PCh. 2.4 - Prob. 94PCh. 2.4 - Prob. 1TFCh. 2.4 - Prob. 2TFCh. 2.4 - Prob. 3TFCh. 2.4 - Prob. 4TFCh. 2.4 - Prob. 5TFCh. 2.4 - Prob. 6TFCh. 2.4 - Prob. 7TFCh. 2.4 - Prob. 8TFCh. 2.4 - Prob. 9TFCh. 2.4 - Prob. 10TFCh. 2.5 - Prob. 1TBCh. 2.5 - Prob. 2TBCh. 2.5 - Prob. 0CCh. 2.5 - 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Prob. 27SCCh. 2 - Prob. 28SCCh. 2 - Prob. 29SCCh. 2 - Prob. 30SCCh. 2 - Prob. 31SCCh. 2 - Prob. 32SCCh. 2 - Prob. 33SCCh. 2 - Prob. 34SCCh. 2 - Prob. 35SCCh. 2 - Prob. 36SCCh. 2 - Prob. 37SCCh. 2 - Prob. 38SCCh. 2 - Prob. 39SCCh. 2 - Prob. 40SCCh. 2 - Prob. 41SCCh. 2 - Prob. 42SCCh. 2 - Prob. 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- show workarrow_forwardA graph of the function f is given below: Study the graph of ƒ at the value given below. Select each of the following that applies for the value a = 1 Of is defined at a. If is not defined at x = a. Of is continuous at x = a. If is discontinuous at x = a. Of is smooth at x = a. Of is not smooth at = a. If has a horizontal tangent line at = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. If has no tangent line at x = a. f(a + h) - f(a) lim is finite. h→0 h f(a + h) - f(a) lim h->0+ and lim h h->0- f(a + h) - f(a) h are infinite. lim does not exist. h→0 f(a+h) - f(a) h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.arrow_forwardThe graph below is the function f(z) 4 3 -2 -1 -1 1 2 3 -3 Consider the function f whose graph is given above. (A) Find the following. If a function value is undefined, enter "undefined". If a limit does not exist, enter "DNE". If a limit can be represented by -∞o or ∞o, then do so. lim f(z) +3 lim f(z) 1-1 lim f(z) f(1) = 2 = -4 = undefined lim f(z) 1 2-1 lim f(z): 2-1+ lim f(x) 2+1 -00 = -2 = DNE f(-1) = -2 lim f(z) = -2 1-4 lim f(z) 2-4° 00 f'(0) f'(2) = = (B) List the value(s) of x for which f(x) is discontinuous. Then list the value(s) of x for which f(x) is left- continuous or right-continuous. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5). If there are none, enter "none". Discontinuous at z = Left-continuous at x = Invalid use of a comma.syntax incomplete. Right-continuous at z = Invalid use of a comma.syntax incomplete. (C) List the value(s) of x for which f(x) is non-differentiable. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5).…arrow_forward
- A graph of the function f is given below: Study the graph of f at the value given below. Select each of the following that applies for the value a = -4. f is defined at = a. f is not defined at 2 = a. If is continuous at x = a. Of is discontinuous at x = a. Of is smooth at x = a. f is not smooth at x = a. If has a horizontal tangent line at x = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. Of has no tangent line at x = a. f(a + h) − f(a) h lim is finite. h→0 f(a + h) - f(a) lim is infinite. h→0 h f(a + h) - f(a) lim does not exist. h→0 h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.arrow_forwardFind the point of diminishing returns (x,y) for the function R(X), where R(x) represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars). R(x) = 10,000-x3 + 42x² + 700x, 0≤x≤20arrow_forwardDifferentiate the following functions. (a) y(x) = x³+6x² -3x+1 (b) f(x)=5x-3x (c) h(x) = sin(2x2)arrow_forward
- x-4 For the function f(x): find f'(x), the third derivative of f, and f(4) (x), the fourth derivative of f. x+7arrow_forwardIn x For the function f(x) = find f'(x). Then find f''(0) and f''(9). 11x'arrow_forwardLet f(x) = √√x+3 and g(x) = 6x − 2. Find each of the following composite functions and state the domain: (a) fog (b) gof, (c) fof (d) gogarrow_forward
- Compute the following: (a) 8x³ + 3x dx (b) cos(2u) du (c) f² ebx dxarrow_forwardFind the following limits. (a) lim 3(x-1)² x→2 x (b) lim 0+x (c) lim 3x2-x+1 x²+3 x²+x-12 x-3 x-3arrow_forwardFor f(x) = (x+3)² - 2 sketch f(x), f(x), f(x − 2), and f(x) — 2. State the coordi- nates of the turning point in each graph.arrow_forward
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