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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.

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02:30

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Oregon State University

Harvey Mudd College

Baylor University

Idaho State University

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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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?

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