{"id":2490,"date":"2018-02-01T15:35:30","date_gmt":"2018-02-01T15:35:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/?post_type=chapter&p=2490"},"modified":"2018-10-03T16:07:47","modified_gmt":"2018-10-03T16:07:47","slug":"chapter-2-review-exercises-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/chapter-2-review-exercises-2\/","title":{"raw":"Chapter 3 Review Exercises","rendered":"Chapter 3 Review Exercises"},"content":{"raw":"\r\n

True or False<\/em>? Justify the answer with a proof or a counterexample.<\/p>\r\n\r\n

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1.\u00a0<\/strong>Every function has a derivative.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738040727\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738040727\"]\r\n

False.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>A continuous function has a continuous derivative.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>A continuous function has a derivative.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738040759\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738040759\"]\r\n

False<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>If a function is differentiable, it is continuous.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Use the limit definition of the derivative to exactly evaluate the derivative.<\/p>\r\n\r\n

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5.\u00a0<\/strong>[latex]f(x)=\\sqrt{x+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738230320\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738230320\"]\r\n

[latex]\\frac{1}{2\\sqrt{x+4}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]f(x)=\\frac{3}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Find the derivatives of the following functions.<\/p>\r\n\r\n

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7.\u00a0<\/strong>[latex]f(x)=3x^3-\\frac{4}{x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738230434\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738230434\"]\r\n

[latex]9x^2+\\frac{8}{x^3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>[latex]f(x)=(4-x^2)^3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]f(x)=e^{\\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738191891\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738191891\"]\r\n

[latex]e^{\\sin x} \\cos x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>[latex]f(x)=\\ln(x+2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11.\u00a0<\/strong>[latex]f(x)=x^2 \\cos x+x \\tan x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738192024\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738192024\"]\r\n

[latex]x \\sec^2 x+2x \\cos x+ \\tan x-x^2 \\sin x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>[latex]f(x)=\\sqrt{3x^2+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>[latex]f(x)=\\frac{x}{4} \\sin^{-1} x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738082332\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738082332\"]\r\n

[latex]\\frac{1}{4}(\\frac{x}{\\sqrt{1-x^2}}+ \\sin^{-1} x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>[latex]x^2 y=(y+2)+xy \\sin (x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Find the following derivatives of various orders.<\/p>\r\n\r\n

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15.\u00a0<\/strong>First derivative of [latex]y=x \\ln x \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738239362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738239362\"]\r\n

[latex] \\cos x (\\ln x+1) -x \\ln x \\sin x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>Third derivative of [latex]y=(3x+2)^2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17.\u00a0<\/strong>Second derivative of [latex]y=4^x+x^2 \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738239502\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738239502\"]\r\n

[latex]4^x(\\ln 4)^2+2 \\sin x+4x \\cos x-x^2 \\sin x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Find the equation of the tangent line to the following equations at the specified point.<\/p>\r\n\r\n

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18.\u00a0<\/strong>[latex]y= \\cos^{-1} x+x[\/latex] at [latex]x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>[latex]y=x+e^x-\\frac{1}{x}[\/latex] at [latex]x=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738226514\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738226514\"]\r\n

[latex]y=(2+e)x-2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Draw the derivative for the following graphs.<\/p>\r\n\r\n

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20.\u00a0<\/strong>\"The<\/span><\/div>\r\n<\/div>\r\n
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21.\u00a0<\/strong>\"The<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1169738226609\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738226609\"]\"The<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by [latex]w(t)=1.9+2.9 \\cos (\\frac{\\pi}{6}t)[\/latex], where [latex]t[\/latex] is measured in hours after midnight, and the height is measured in feet.<\/p>\r\n\r\n

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22.\u00a0<\/strong>Find and graph the derivative. What is the physical meaning?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>Find [latex]w^{\\prime}(3)[\/latex]. What is the physical meaning of this value?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738138131\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738138131\"]\r\n

[latex]w^{\\prime}(3)=-\\frac{2.9\\pi}{6}[\/latex]. At 3 a.m. the tide is decreasing at a rate of 1.514 ft\/hr.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Wind Speeds of Hurricane Katrina\r\nSource: http:\/\/news.nationalgeographic.com\/news\/2005\/09\/0914_050914_katrina_timeline.html.<\/caption>\r\n
Hours after Midnight, August 26<\/th>\r\nWind Speed (mph)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1<\/td>\r\n45<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n75<\/td>\r\n<\/tr>\r\n
11<\/td>\r\n100<\/td>\r\n<\/tr>\r\n
29<\/td>\r\n115<\/td>\r\n<\/tr>\r\n
49<\/td>\r\n145<\/td>\r\n<\/tr>\r\n
58<\/td>\r\n175<\/td>\r\n<\/tr>\r\n
73<\/td>\r\n155<\/td>\r\n<\/tr>\r\n
81<\/td>\r\n125<\/td>\r\n<\/tr>\r\n
85<\/td>\r\n95<\/td>\r\n<\/tr>\r\n
107<\/td>\r\n35<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
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24.\u00a0<\/strong>Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>Estimate the derivative of the wind speed at hour 83. What is the physical meaning?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1169738074898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738074898\"]\r\n

-7.5. The wind speed is decreasing at a rate of 7.5 mph\/hr<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

True or False<\/em>? Justify the answer with a proof or a counterexample.<\/p>\n

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\n

1.\u00a0<\/strong>Every function has a derivative.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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False.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2.\u00a0<\/strong>A continuous function has a continuous derivative.<\/p>\n<\/div>\n<\/div>\n

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3.\u00a0<\/strong>A continuous function has a derivative.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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False<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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4.\u00a0<\/strong>If a function is differentiable, it is continuous.<\/p>\n<\/div>\n<\/div>\n

Use the limit definition of the derivative to exactly evaluate the derivative.<\/p>\n

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5.\u00a0<\/strong>[latex]f(x)=\\sqrt{x+4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\frac{1}{2\\sqrt{x+4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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6.\u00a0<\/strong>[latex]f(x)=\\frac{3}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

Find the derivatives of the following functions.<\/p>\n

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7.\u00a0<\/strong>[latex]f(x)=3x^3-\\frac{4}{x^2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]9x^2+\\frac{8}{x^3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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8.\u00a0<\/strong>[latex]f(x)=(4-x^2)^3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9.\u00a0<\/strong>[latex]f(x)=e^{\\sin x}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]e^{\\sin x} \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10.\u00a0<\/strong>[latex]f(x)=\\ln(x+2)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11.\u00a0<\/strong>[latex]f(x)=x^2 \\cos x+x \\tan x[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x \\sec^2 x+2x \\cos x+ \\tan x-x^2 \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12.\u00a0<\/strong>[latex]f(x)=\\sqrt{3x^2+2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13.\u00a0<\/strong>[latex]f(x)=\\frac{x}{4} \\sin^{-1} x[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\frac{1}{4}(\\frac{x}{\\sqrt{1-x^2}}+ \\sin^{-1} x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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14.\u00a0<\/strong>[latex]x^2 y=(y+2)+xy \\sin (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

Find the following derivatives of various orders.<\/p>\n

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15.\u00a0<\/strong>First derivative of [latex]y=x \\ln x \\cos x[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\cos x (\\ln x+1) -x \\ln x \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16.\u00a0<\/strong>Third derivative of [latex]y=(3x+2)^2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17.\u00a0<\/strong>Second derivative of [latex]y=4^x+x^2 \\sin x[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]4^x(\\ln 4)^2+2 \\sin x+4x \\cos x-x^2 \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Find the equation of the tangent line to the following equations at the specified point.<\/p>\n

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18.\u00a0<\/strong>[latex]y= \\cos^{-1} x+x[\/latex] at [latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19.\u00a0<\/strong>[latex]y=x+e^x-\\frac{1}{x}[\/latex] at [latex]x=1[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]y=(2+e)x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Draw the derivative for the following graphs.<\/p>\n

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20.\u00a0<\/strong>\"The<\/span><\/div>\n<\/div>\n
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21.\u00a0<\/strong>\"The<\/span><\/div>\n
\n
Show Solution<\/span><\/p>\n
\"The<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by [latex]w(t)=1.9+2.9 \\cos (\\frac{\\pi}{6}t)[\/latex], where [latex]t[\/latex] is measured in hours after midnight, and the height is measured in feet.<\/p>\n

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22.\u00a0<\/strong>Find and graph the derivative. What is the physical meaning?<\/p>\n<\/div>\n<\/div>\n

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23.\u00a0<\/strong>Find [latex]w^{\\prime}(3)[\/latex]. What is the physical meaning of this value?<\/p>\n<\/div>\n

\n
Show Solution<\/span><\/p>\n
\n

[latex]w^{\\prime}(3)=-\\frac{2.9\\pi}{6}[\/latex]. At 3 a.m. the tide is decreasing at a rate of 1.514 ft\/hr.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Wind Speeds of Hurricane Katrina
\nSource: http:\/\/news.nationalgeographic.com\/news\/2005\/09\/0914_050914_katrina_timeline.html.<\/caption>\n
Hours after Midnight, August 26<\/th>\nWind Speed (mph)<\/th>\n<\/tr>\n<\/thead>\n
1<\/td>\n45<\/td>\n<\/tr>\n
5<\/td>\n75<\/td>\n<\/tr>\n
11<\/td>\n100<\/td>\n<\/tr>\n
29<\/td>\n115<\/td>\n<\/tr>\n
49<\/td>\n145<\/td>\n<\/tr>\n
58<\/td>\n175<\/td>\n<\/tr>\n
73<\/td>\n155<\/td>\n<\/tr>\n
81<\/td>\n125<\/td>\n<\/tr>\n
85<\/td>\n95<\/td>\n<\/tr>\n
107<\/td>\n35<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
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24.\u00a0<\/strong>Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?<\/p>\n<\/div>\n<\/div>\n

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25.\u00a0<\/strong>Estimate the derivative of the wind speed at hour 83. What is the physical meaning?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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-7.5. The wind speed is decreasing at a rate of 7.5 mph\/hr<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Shared previously<\/div>