Linearization: Tangent Planes and Differentials

1. Find the equation of the tangent plane to $f(x,y) = x^2y$ at the point $(1,2)$.

2. Suppose that the tangent plane to the graph of a continuously differentiable function $z=g(x,y)$ is given in the form . Use the equation of the tangent plane to identify a point on the graph as well as a value of $g_x$ and a value of $g_y$. Be sure to identify at what point(s) you have found the values of the partial derivatives.

1. Suppose that the elevation of a plot of land is given by the function $h$, where we additionally know that $h(3,1) = 4.35$, $h_x(3,1) = 0.27$, and $h_y(3,1) = -0.19$. Assume that $x$ and $y$ are measured in miles in the east and north directions, respectively, from $(0,0)$.

   Your GPS device says that you are currently at the point $(3,1)$. However, you know that the coordinates are only accurate to within $0.2$ miles; that is, $dx = \Delta x = 0.2$ and $dy= \Delta y = 0.2$. Estimate the uncertainty in you…

Let $f$ be the function defined by $f(x,y) = 2x^2+3y^3$.

1. Find the equation of the tangent plane to $f$ at the point $(1,2)$.

2. Use the linearization to approximate the values of $f$ at the points $(1.1, 2.05)$ and $(1.3,2.2)$.

3. Compare the approximations form part (b) to the exact values of $f(1.1, 2.05)$ and $f(1.3, 2.2)$. Which approximation is more accurate. Explain why this should be expected.

Let $g$ be a function that is differentiable at $(-2,5)$ and suppose that its tangent plane at this point is given by $z = -7 + 4(x+2) - 3(y-5)$.

1. Determine the values of $g(-2,5)$, $g_x(-2,5)$, and $g_y(-2,5)$. Write one sentence to explain your thinking.

2. Estimate the value of $g(-1.8, 4.7)$. Clearly show your work and thinking.

3. Given changes of $dx = -0.34$ and $dy = 0.21$, estimate the corresponding change in $g$ that is given by its differential, $dg$.

4. Suppose that another function $h$ is also diffe…

In the following questions, we determine and apply the linearization for several different functions.

1. Find the linearization $L(x,y)$ for the function $f$ defined by $f(x,y) = \cos(x)(2e^{2y}+e^{-2y})$ at the point $(x_0,y_0) = (0,0)$. Use the linearization to estimate the value of $f(0.1, 0.2)$. Compare your estimate to the actual value of $f(0.1, 0.2)$.

2. The Heat Index, $I$, (measured in apparent degrees F) is a function of the actual temperature $T$ outside (in degrees F) and the rela…

In the following questions, we investigate two different applied settings using the differential.

1. Let $f$ represent the vertical displacement in centimeters from the rest position of a string (like a guitar string) as a function of the distance $x$ in centimeters from the fixed left end of the string and $y$ the time in seconds after the string has been plucked. (An interesting video of this can be seen…

We know that if a function of a single variable is differentiable at a point, then that function is also continuous at that point. In this exercise we determine that the same property holds for functions of two variables. A function $f$ of the two variables $x$ and $y$ is continuous at a point $(x_0,y_0)$ in its domain if or (letting $x=x_0+h$ and $y = y_0 + k$, Show that if $f$ is differentiable at $(x_0,y_0)$, then $f$ is continuous at $(x_0,y_0)$. (Hi…