The derivative plays a central role in first semester calculus because it provides important information about a function. Thinking graphically, for instance, the derivative at a point tells us the slope of the tangent line to the graph at that point. In addition, the derivative at a point also provides the instantaneous rate of change of the function with respect to changes in the independent variable.
Now that we are investigating functions of two or more variables, we can still ask how fast the function is changing, though we have to be careful about what we mean. Thinking graphically again, we can try to measure how steep the graph of the function is in a particular direction. Alternatively, we may want to know how fast a function’s output changes in response to a change in one of the inputs. Over the next few sections, we will develop tools for addressing issues such as these. Preview Activity 10.2.1 explores some issues with what we will come to call partial derivatives.
Suppose the interest rate is fixed at . Express as a function of alone using . That is, let . Sketch the graph of on the left of Figure 10.2.1. Explain the meaning of the function .
Find the instantaneous rate of change and state the units on this quantity. What information does tell us about our car loan? What information does tell us about the graph you sketched in (b)?
Express as a function of alone, using a fixed time of . That is, let . Sketch the graph of on the right of Figure 10.2.1. Explain the meaning of the function .
Find the instantaneous rate of change and state the units on this quantity. What information does tell us about our car loan? What information does tell us about the graph you sketched in (d)?
In Section 9.1, we studied the behavior of a function of two or more variables by considering the traces of the function. Recall that in one example, we considered the function defined by
which measures the range, or horizontal distance, in feet, traveled by a projectile launched with an initial speed of feet per second at an angle radians to the horizontal. The graph of this function is given again on the left in Figure 10.2.2. Moreover, if we fix the angle , we may view the trace as a function of alone, as seen at right in Figure 10.2.2.
which gives the slope of the tangent line shown on the right of Figure 10.2.2. Thinking of this derivative as an instantaneous rate of change implies that if we increase the initial speed of the projectile by one foot per second, we expect the horizontal distance traveled to increase by approximately 8.74 feet if we hold the launch angle constant at radians.
By holding fixed and differentiating with respect to , we obtain the first-order partial derivative of with respect to . Denoting this partial derivative as , we have seen that
Once again, the derivative gives the slope of the tangent line shown on the right in Figure 10.2.3. Thinking of the derivative as an instantaneous rate of change, we expect that the range of the projectile increases by 509.5 feet for every radian we increase the launch angle if we keep the initial speed of the projectile constant at 150 feet per second.
By holding fixed and differentiating with respect to , we obtain the first-order partial derivative of with respect to . As before, we denote this partial derivative as and write
As with the partial derivative with respect to , we may express this quantity more generally at an arbitrary point . To recap, we have now arrived at the formal definition of the first-order partial derivatives of a function of two variables.
Write the trace at the fixed value . On the left side of Figure 10.2.5, draw the graph of the trace with around the point where , indicating the scale and labels on the axes. Also, sketch the tangent line at the point .
Write the trace at the fixed value . On the right side of Figure 10.2.5, draw the graph of the trace with indicating the scale and labels on the axes. Also, sketch the tangent line at the point .
As these examples show, each partial derivative at a point arises as the derivative of a one-variable function defined by fixing one of the coordinates. In addition, we may consider each partial derivative as defining a new function of the point , just as the derivative defines a new function of in single-variable calculus. Due to the connection between one-variable derivatives and partial derivatives, we will often use Leibniz-style notation to denote partial derivatives by writing
To see the contrast between how we calculate single variable derivatives and partial derivatives, and the difference between the notations and , observe that
Thus, computing partial derivatives is straightforward: we use the standard rules of single variable calculus, but do so while holding one (or more) of the variables constant.
Subsection10.2.2Interpretations of First-Order Partial Derivatives
Recall that the derivative of a single variable function has a geometric interpretation as the slope of the line tangent to the graph at a given point. Similarly, we have seen that the partial derivatives measure the slope of a line tangent to a trace of a function of two variables as shown in Figure 10.2.6.
Now we consider the first-order partial derivatives in context. Recall that the difference quotient for a function of a single variable at a point where tells us the average rate of change of over the interval , while the derivative tells us the instantaneous rate of change of at . We can use these same concepts to explain the meanings of the partial derivatives in context.
Here is the speed of sound in meters/second, is the temperature in degrees Celsius, is the salinity in grams/liter of water, and is the depth below the ocean surface in meters.
State the units in which each of the partial derivatives, , and , are expressed and explain the physical meaning of each.
Evaluate each of the three partial derivatives at the point where , and . What does the sign of each partial derivatives tell us about the behavior of the function at the point ?
Subsection10.2.3Using tables and contours to estimate partial derivatives
Remember that functions of two variables are often represented as either a table of data or a contour plot. In single variable calculus, we saw how we can use the difference quotient to approximate derivatives if, instead of an algebraic formula, we only know the value of the function at a few points. The same idea applies to partial derivatives.
The wind chill, as frequently reported, is a measure of how cold it feels outside when the wind is blowing. In Table 10.2.7, the wind chill , measured in degrees Fahrenheit, is a function of the wind speed , measured in miles per hour, and the ambient air temperature , also measured in degrees Fahrenheit. We thus view as being of the form .
Estimate the partial derivative . What are the units on this quantity and what does it mean? (Recall that we can estimate a partial derivative of a single variable function using the symmetric difference quotient for small values of . A partial derivative is a derivative of an appropriate trace.)
Shown below in Figure 10.2.8 is a contour plot of a function . The values of the function on a few of the contours are indicated to the left of the figure.
Suppose you have a different function , and you know that ,, and . Using this information, sketch a possibility for the contour passing through on the left side of Figure 10.2.9. Then include possible contours and .
Suppose you have yet another function , and you know that ,, and . Using this information, sketch a possible contour passing through on the right side of Figure 10.2.9. Then include possible contours and .
The partial derivative tells us the instantaneous rate of change of with respect to at when is fixed at . Geometrically, the partial derivative tells us the slope of the line tangent to the trace of the function at the point .
The partial derivative tells us the instantaneous rate of change of with respect to at when is fixed at . Geometrically, the partial derivative tells us the slope of the line tangent to the trace of the function at the point .
Suppose that is a smooth function and that its partial derivatives have the values, and . Given that , use this information to estimate the value of . Note this is analogous to finding the tangent line approximation to a function of one variable. In fancy terms, it is the first Taylor approximation.
Determine the sign of and at each indicated point using the contour diagram of shown below. (The point is that in the first quadrant, at a positive and value; through are located clockwise from , so that is at a positive value and negative , etc.)
Your monthly car payment in dollars is , where $ is the amount you borrowed, is the number of months it takes to pay off the loan, and percent is the interest rate.
(For this problem, write our your units in full, writing dollars for $, months for months, percent for %, etc. Note that fractional units generally have a plural numerator and singular denominator.)
(For this problem, write our your units in full, writing dollars for $, months for months, percent for %, etc. Note that fractional units generally have a plural numerator and singular denominator.)
An experiment to measure the toxicity of formaldehyde yielded the data in the table below. The values show the percent, , of rats surviving an exposure to formaldehyde at a concentration of (in parts per million, ppm) after months.
An airport can be cleared of fog by heating the air. The amount of heat required depends on the air temperature and the wetness of the fog. The figure below shows the heat required (in calories per cubic meter of fog) as a function of the temperature (in degrees Celsius) and the water content (in grams per cubic meter of fog). Note that this figure is not a contour diagram, but shows cross-sections of with fixed at ,,, and .
The Heat Index, , (measured in apparent degrees F) is a function of the actual temperature outside (in degrees F) and the relative humidity (measured as a percentage). A portion of the table which gives values for this function, , is reproduced in Table 10.2.10.
State the limit definition of the value . Then, estimate , and write one complete sentence that carefully explains the meaning of this value, including its units.
State the limit definition of the value . Then, estimate , and write one complete sentence that carefully explains the meaning of this value, including its units.
On a certain day, at 1 p.m. the temperature is 92 degrees and the relative humidity is 85%. At 3 p.m., the temperature is 96 degrees and the relative humidity 75%. What is the average rate of change of the heat index over this time period, and what are the units on your answer? Write a sentence to explain your thinking.
Often we are given certain graphical information about a function instead of a rule. We can use that information to approximate partial derivatives. For example, suppose that we are given a contour plot of the kinetic energy function (as in Figure 10.2.11) instead of a formula. Use this contour plot to approximate and as best you can. Compare to your calculations from earlier parts of this exercise.
Assume that temperature is measured in degrees Celsius and that and are each measured in inches. (Note: At no point in the following questions should you expand the denominator of .)
If an ant is on the metal plate, standing at the point , and starts walking in the direction parallel to the positive axis, at what rate will the temperature the ant is experiencing change? Explain, and include appropriate units.
If an ant is walking along the line in the positive direction, at what instantaneous rate will the temperature the ant is experiencing change when the ant passes the point ?
Now suppose the ant is stationed at the point and walks in a straight line towards the point . Determine the average rate of change in temperature (per unit distance traveled) the ant encounters in moving between these two points. Explain your reasoning carefully. What are the units on your answer?
Determine the equation of the plane that passes through the point whose normal vector is orthogonal to the direction vectors of the two lines found in (b) and (c).
Recall from single variable calculus that, given the derivative of a single variable function and an initial condition, we can integrate to find the original function. We can sometimes use the same process for functions of more than one variable. For example, suppose that a function satisfies ,, and .
Find all possible functions of and such that . Your function will have both and as independent variables and may also contain summands that are functions of alone.