The figure shows the temperature values (in °F) on a typical May day in a certain Midwestern city. t (x) degrees F 80 70 60 50 40 x hours since 6 A.M. 2 4 6 8 10 12 The equation of the graph is t(x) = -0.8x^2 + 11.6x + 38.2°F where x is the number of hours since 6 A.M. (a) Write the formula for t'. t'(x) = (b) How quickly is the temperature changing at 11 A.M.? (Round your answer to one decimal place.) t'(5) = °F per hour (c) What is the instantaneous rate of change of the temperature at 4 P.M.? (Round your answer to one decimal place.) t'(10) = °F per hour
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8x^2 + 11.6x + 38.2 (a) We are asked to find the formula for the rate of change of temperature with respect to time. This is the derivative of the temperature function with respect to x. Show more…
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Midwest Temperature The figure shows the temperature values (in ${ }^{\circ} \mathrm{F}$ ) on a typical May day in a certain Midwestern city.The equation of the graph is $$ t(x)=-0.8 x^{2}+11.6 x+38.2^{\circ} \mathrm{F} $$ where $x$ is the number of hours since 6 A.M. a. Write the formula for $t^{\prime}$. b. How quickly is the temperature changing at $10 \mathrm{~A} . \mathrm{M} . ?$ c. What is the instantaneous rate of change of the temperature at $4 \mathrm{PM} . ?$ d. Draw and label tangent lines depicting the results from parts $b$ and $c$
Determining Change: Derivatives
Simple Rate-of-Change Formulas
The temperature, T, in degrees Fahrenheit t hours after 6 AM is given by: T(t) = -t^2 + 8t + 32. Find and interpret T(4), T(8), and T(12). Find and interpret the average rate of change of the temperature between 10 AM and 6 PM. Find and simplify a formula for the average rate of change of T from 12 to 12 + h. Find the limit of your answer to part (e) as h approaches 0. What does your answer mean in this applied situation? Sketch a careful graph of y = T(t) and geometrically interpret your answers.
Brent B.
Temperature The given graph shows the temperature $T$ in $^{\circ} \mathrm{F}$ at Davis, $\mathrm{CA},$ on April $18,2008,$ between 6 A.M. and 6 $\mathrm{em}.$ a. Estimate the rate of temperature change at the times $$7 \text { A.M. } \quad \text { ii) } 9 \text { A.M. } \quad \text { iii) } 2 \text { P.M. } \quad \text { iv } ) 4 $$ b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times? c. Use the graphical technique of Example 3 to graph the derivative of temperature $T$ versus time $t$.
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The Derivative as a Function
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