Problem 11
Question
In Problems 11-14, find the gradient vector of the given function at the given point \(\mathbf{p}\). Then find the equation of the tangent plane at \(\mathbf{p}\) (see Example 1). f(x, y)=x^{2} y-x y^{2}, \mathbf{p}=(-2,3)
Step-by-Step Solution
Verified Answer
Gradient vector: \( \langle -21, 16 \rangle \), Tangent plane: \( z = -21x + 16y - 108 \).
1Step 1: Find Partial Derivatives
First, find the partial derivatives of the function \( f(x, y) = x^2 y - x y^2 \) with respect to \( x \) and \( y \). For \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant:\[ \frac{\partial f}{\partial x} = 2x y - y^2 \]For \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant:\[ \frac{\partial f}{\partial y} = x^2 - 2x y \]
2Step 2: Evaluate Partial Derivatives at Point \( \mathbf{p} \)
Substitute the point \( \mathbf{p} = (-2, 3) \) into the partial derivatives to find the gradient vector. \[ \frac{\partial f}{\partial x}(-2, 3) = 2(-2)(3) - 3^2 = -12 - 9 = -21 \]\[ \frac{\partial f}{\partial y}(-2, 3) = (-2)^2 - 2(-2)(3) = 4 + 12 = 16 \]
3Step 3: Form the Gradient Vector
The gradient vector is composed of the partial derivatives evaluated at the given point.So, the gradient vector \( abla f(-2, 3) \) is:\[ abla f(-2, 3) = \langle -21, 16 \rangle \]
4Step 4: Find the Equation of the Tangent Plane
The equation of the tangent plane at a point \( (a, b) \) on a function \( f(x, y) \) with gradient \( abla f(a, b) = \langle f_x(a, b), f_y(a, b) \rangle \) is:\[ z - f(a, b) = f_x(a, b)(x - a) + f_y(a, b)(y - b) \] Evaluate \( f(x, y) \) at \( \mathbf{p} = (-2, 3) \):\[ f(-2, 3) = (-2)^2(3) - (-2)(3)^2 = 12 + 18 = -18 \]Substitute \( a = -2, b = 3, f(-2, 3) = -18, f_x(-2, 3) = -21, f_y(-2, 3) = 16 \):\[ z + 18 = -21(x + 2) + 16(y - 3) \] Simplify:\[ z + 18 = -21x - 42 + 16y - 48 \]\[ z = -21x + 16y - 108 \]
5Step 5: Final Equation of the Tangent Plane
The final equation for the tangent plane of the function \( f(x, y) \) at the point \( \mathbf{p} = (-2, 3) \) is:\[ z = -21x + 16y - 108 \]
Key Concepts
Partial DerivativesTangent PlaneMultivariable CalculusDifferential Calculus
Partial Derivatives
Partial derivatives are a key concept in calculus, especially when dealing with functions of multiple variables. They tell us how a function changes as we tweak one variable while keeping others fixed.
In this exercise, we calculate the partial derivative of the function \( f(x, y) = x^2 y - x y^2 \) with respect to each variable, treating all other variables as constants. For example:
In this exercise, we calculate the partial derivative of the function \( f(x, y) = x^2 y - x y^2 \) with respect to each variable, treating all other variables as constants. For example:
- The partial derivative with respect to \( x \), denoted \( \frac{\partial f}{\partial x} \), considers \( y \) as a constant, resulting in \( 2xy - y^2 \).
- The partial derivative with respect to \( y \), denoted \( \frac{\partial f}{\partial y} \), sees \( x \) as a constant, leading to \( x^2 - 2xy \).
Tangent Plane
Imagine slicing through a 3D surface with a flat sheet. This flat piece is akin to what we call the tangent plane. It gives us a linear approximation of the surface near a specific point.
The concept of a tangent plane is essential when working with surfaces in three-dimensional space, and here, it's used to find the tangent to our function \( f(x, y) \) at point \( \mathbf{p} = (-2,3) \).
We utilize the gradient vector, found by evaluating partial derivatives at our given point, to form the equation of the tangent plane. The result is a plane that gives linear estimates of the function's value in the immediate vicinity of the point. This plane helps visualize and understand how the function behaves locally.
The concept of a tangent plane is essential when working with surfaces in three-dimensional space, and here, it's used to find the tangent to our function \( f(x, y) \) at point \( \mathbf{p} = (-2,3) \).
We utilize the gradient vector, found by evaluating partial derivatives at our given point, to form the equation of the tangent plane. The result is a plane that gives linear estimates of the function's value in the immediate vicinity of the point. This plane helps visualize and understand how the function behaves locally.
Multivariable Calculus
Multivariable calculus expands the principles of single-variable calculus into higher dimensions, where functions depend on multiple variables. This branch of mathematics is essential for understanding and solving complex problems found in fields like physics and engineering.
In this exercise, we work with a function, \( f(x, y) \), which is dependent on two variables. The methods used, such as the calculation of partial derivatives and the evaluation of the tangent plane, are core techniques in multivariable calculus.
The gradient vector, derived from these partial derivatives, serves as a fundamental tool in multivariable functions, highlighting the direction of steepest ascent.
By mastering these concepts, students can transition from theoretical understanding to practical application in real-world scenarios, analyzing situations where multiple factors change simultaneously.
In this exercise, we work with a function, \( f(x, y) \), which is dependent on two variables. The methods used, such as the calculation of partial derivatives and the evaluation of the tangent plane, are core techniques in multivariable calculus.
The gradient vector, derived from these partial derivatives, serves as a fundamental tool in multivariable functions, highlighting the direction of steepest ascent.
By mastering these concepts, students can transition from theoretical understanding to practical application in real-world scenarios, analyzing situations where multiple factors change simultaneously.
Differential Calculus
Differential calculus is the study of how quantities change and is fundamentally linked to the process of differentiation, which allows us to find rates of change and slopes of curves.
In this exercise, the function \( f(x, y) \) is differentiated with respect to its variables to obtain partial derivatives. These derivatives provide insight into how the function changes as \( x \) and \( y \) vary separately.
Whether dealing with the slopes of lines or the curves of more complex surfaces, differential calculus offers the tools to break down these changes into more manageable pieces.
This form of calculus is not only about calculating derivatives but also using them creatively to solve problems, like determining the equation of tangent planes, which, in turn, offer a deeper understanding of the geometry and behavior of multivariable functions.
In this exercise, the function \( f(x, y) \) is differentiated with respect to its variables to obtain partial derivatives. These derivatives provide insight into how the function changes as \( x \) and \( y \) vary separately.
Whether dealing with the slopes of lines or the curves of more complex surfaces, differential calculus offers the tools to break down these changes into more manageable pieces.
This form of calculus is not only about calculating derivatives but also using them creatively to solve problems, like determining the equation of tangent planes, which, in turn, offer a deeper understanding of the geometry and behavior of multivariable functions.
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