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Let $ H(t) $ be the daily cost (in dollars) to heat an office building when the outside temperature is $ t $ degrees Fahrenheit.

(a) What is the meaning of $ H'(58) $? What are its units?

(b) Would you expect $ H'(58) $ to be positive or negative? Explain.

(a) $H^{\prime}(58)$ is the rate at which the daily heating cost changes with respect to temperature when the outside temperature is $58^{\circ} \mathrm{F}$. The units are dollars/ $^{\circ} \mathrm{F}$.

(b) If the outside temperature increases, the building should require less heating, so we would expect $H^{\prime}(58)$ to be negative.

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he's Fahrenheit. H. Is the cost or the dollars. The number of dollars that's going to cost to heat a building. So aged a number of dollars to heat a building is a function depending on the variable T. To temperature in degrees Fahrenheit. Each prime of T. The derivative of H. With respect to T. How does the cost to heat the building change with respect to a change in temperature. If each prime of T. But let me uh restate that if H. Prime of 58 Let's investigate h. prime of 58. Now the 58 Is in the location of the T. variable. So this means the temperature outside is 58°. Each prime of 58 means the rate at which the cost to heat the building is going to change with respect to an increase in temperature above 58°. Okay think of H. Prime the derivative of H. With respect to T. As being to change in H. Over to change into. So how does h the cost to heat the building Change with respect to a change in the temperature. So once again h. prime of 58 as the temperature increases slightly above 58° H. prime of 58 is going to be the increase in the cost to heat the building. So as the temperature rises above 58°, how does the heating cost change with respect to the change in the temperature? Now If the temperature is increasing above 58°, that means it's getting warmer outside, you're not going to have to heat uh the building uh as much So the heating cost should start to go down or decrease. So as the temperature increases, the rate at which it cost you to heat the building should decrease. So we would expect h. prime of 58 to be negative or less than zero. That's because as the temperature outside is increasing the cost to heat the building should be going down And that's why uh h prime of 58 should be negative less than zero. What would the units be? Uh The units for H. Prime of 58 units would be simple. Uh dollars, her degrees Fahrenheit. So how how many dollars to heat the building? Uh how is the number of dollars to heat the building changing with respect to a change in 2°F. So this would be your unit of mission?