3.5 Second-Order Partial Derivatives

Second-Order Partial Derivatives

Introduction

Second-Order Partial Derivatives

Interpreting the Second-Order Partial Derivatives

Once again, we consider the function $f$ defined by $f(x,y) = \frac{x^2\sin(2y)}{32}$ that measures a projectile's range as a function of its initial speed $x$ and launch angle $y$. The graph of this function, including traces with $x=150$ and $y=0.6$, is shown in the interactive graph below.

Compute the partial derivatives $f_x$ and $f_y$ as functions of $x$ and $y$.

Work through this exercise and explain your reasoning step by step.

If $h(x,y,z,t) = 9x^9z-xyz^9 + 9t$, how many second-order partial derivatives does the function $h$ have? Write a sentence to justify your reasoning on the number of second-order partial derivatives of $h$. Finally, find $h_{xz}$ and $h_{zx}$ (you do not need to find the other second-order partial derivatives).

Work through this exercise and explain your reasoning step by step.

In the interactive graphic below we see the trace of $f(x,y) = \sin(x) e^{-y}$ with $x$ held constant with $x = 1.75$ plotted in blue. Use the slider to investigate how the slope of the tangent line changes as you vary the $y$-coordinate along this trace. Write a couple of sentences that describe whether the slope of the tangent lines to this curve increase or decrease as $y$ increases along the trace $x=1.75$. Be sure to pay attention to which direction corresponds to each coordinate increasing.

Compute $f_{yy}(x,y)$ algebraically and explain how your observations in the previous part are related to the value of $f_{yy}(1.75,y)$. Your response should address the notion of concavity. Be careful to note the directions in which $y$ is increasing.

We want to explore what the mixed partial derivative $f_{xy}$ describes. We will do this by focusing on the point $(x,y)=(1.75,-1.5)$. In this part, we will work through the definition for $f_{xy} (1.75,-1.5)$ carefully.

The partial derivative $f_x(1.75,-1.5)$ measures the slope of the line tangent to the trace given by $y=-1.5$. These tangent lines change in the $x$-direction (parallel to the $x$-axis). When we consider $f_{xy}$, we are taking the partial derivative of $f_x$ with respect to $y$, $\frac{\partial}{\partial y} \Bigl[ f_x \Bigr]$. Thus, we are considering how the slope of the tangent line in the $x$-direction changes when we vary the $y$-coordinate a small amount. We can approximate $f_{xy}(1.75,-1.5)$ with the difference quotient

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One way of visualizing this is by thinking of sliding a pencil along the trace with $x=1.5$ so that the pencil is in the $x$-direction and tangent to the surface. In the interactive image below, the pencil would be the black tangent line drawn. Use the slider at the top of the interactive to change the $y$-coordinate of the point where the tangent line is drawn. Examine what happens to the slope of the tangent line as you increase the $y$-coordinate of the point of tangency by adjusting the slider. The tangent line in the $x$-direction at the point $(1.75,-1.5)$ is drawn in gray for reference purposes.

Based on your exploration, write a few sentences about whether $f_{xy}(1.75, -1.5)$ is positive or negative and justify your reasoning.

Compute $f_{xy}(x,y)$ algebraically and evaluate $f_{xy}(1.75, -1.5)$. Write a couple of sentences about how this value compares with your observations in the previous part.

We know that $f_{xx}(1.75, -1.5)$ measures the concavity of the $y = -1.5$ trace, and that $f_{yy}(1.75, -1.5)$ measures the concavity of the $x = 1.75$ trace. What do you think the quantity $f_{xy}(1.75, -1.5)$ measures?

In the interactive image below, the trace with $y = -1.5$ is highlighted with the point $(1.75,-1.5,f(1.75,-1.5))$ drawn in black. Sketch three tangent lines to this trace whose slopes correspond to the value of $f_{yx}(x,-1.5)$ for three different values of $x$ near $x=1.75$. Use your tangent lines to state whether $f_{yx}(1.75, -1.5)$ is positive or negative. Justify your reasoning and describe what you think $f_{yx}(1.75, -1.5)$ measures.

Estimate the partial derivatives $w_{T}(20,-15)$, $w_{T}(20,-10)$, and $w_T(20,-5)$. Use these results to estimate the second-order partial $w_{TT}(20, -10)$.

In a similar way, estimate the second-order partial $w_{vv}(20,-10)$.

Estimate the partial derivatives $w_T(20,-10)$, $w_T(25,-10)$, and $w_T(15,-10)$, and use your results to estimate the partial $w_{Tv}(20,-10)$.

In a similar way, estimate the partial derivative $w_{vT}(20,-10)$.

Write a few sentences to explain what the values $w_{TT}(20, -10)$, $w_{vv}(20,-10)$, and $w_{Tv}(20,-10)$ indicate regarding the behavior of $w(v,T)$.