Graphical and intuitive understanding of derivatives and integrals 1. The temperature of ocean water drops as the ocean depth increases. Thus, temperature is a function of (depends on) depth. This relationship becomes more dramatic at deeper parts of the ocean, where sunlight cannot penetrate. Mathematically, we write this as T(x), where x is depth (meters), and T is temperature (deg. C). a. Label the independent and dependent variable on this figure by filling in the corresponding circle in each box (Source: US National Weather Service; = SFC surface). Hint: Consider the details given about the relationship SFC Epipelagic Zone 200 400 mperature Profil 600 Thermocline Independent O Dependent 800 1000 1200 2 6 8 10 12 14 Temperature (C) b. What is unique about the orientation of this figure? Why do you think the authors chose to display the relationship this way? Independent Dependent c. What is the value of T(0)? Approximate by visually interpreting the figure. Report 3 significant figures and include units d. What would the equation for T'(x) tell you (if you had the equation)? (check all that apply) The total temperature accumulation with ocean depth change O The first derivative of T(x) O The integral of T(x) O The change in temperature per m O The change in temperature per m per m O The change in temperature per unit change in ocean depth e. What are the units of T'(x)? f. Do you expect T'(x) to be positive or negative? Why? O Positive O Negative Briefly, explain why: 1