Problem 3 - Advanced Engineering Mathematics

Question

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=y-x^{2} ;(2,5) $$

Step-by-Step Solution

Answer

The level curve is a parabola, \(y = x^2 + 1\), and the gradient at (2, 5) is \((-4, 1)\).

Step 1: Understand the Level Curve

A level curve of a function of two variables, like \( f(x, y) = y - x^2 \), is a curve along which the function has a constant value. For the level curve passing through point \((2, 5)\), we first need to evaluate the function at this point.

Step 2: Calculate the Function Value at the Point

Substitute the coordinates of the point \((2, 5)\) into the function \(f(x, y) = y - x^2\):\[f(2, 5) = 5 - (2)^2 = 5 - 4 = 1.\]The level curve is where \(f(x, y) = 1\).

Step 3: Find the Equation of the Level Curve

The level curve can be described by setting the expression \(y - x^2\) equal to the constant value \(1\):\[y - x^2 = 1.\]Rearraging this gives:\[y = x^2 + 1.\]

Step 4: Sketch the Level Curve

The equation \(y = x^2 + 1\) represents a parabola that opens upwards with its vertex at \((0, 1)\). Sketch this parabola on the coordinate plane. Ensure it passes through the point \((2, 5)\).

Step 5: Calculate the Gradient

The gradient of a function \(f(x, y)\) is given by the vector of partial derivatives:\(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right).\)For \(f(x, y) = y - x^2\), the partial derivatives are:\(\frac{\partial f}{\partial x} = -2x,\)\(\frac{\partial f}{\partial y} = 1.\)

Step 6: Evaluate the Gradient at the Point

Substitute \((2, 5)\) into the gradient:\[abla f(2, 5) = (-2 \cdot 2, 1) = (-4, 1).\]This is the direction of steepest ascent from the point \((2, 5)\).

Step 7: Sketch the Gradient Vector

Draw the gradient vector \((-4, 1)\) starting at the point \((2, 5)\). This vector should point in the direction \(-4\) units in the x-direction and \(+1\) unit in the y-direction.

Key Concepts

Gradient

The concept of a gradient is fundamental in understanding how a function changes at any point. In simple terms, the gradient of a function at a given point indicates the direction of the steepest ascent. It's a vector composed of partial derivatives.

The gradient for the function \(f(x, y) = y - x^2\) is found by calculating its partial derivatives. For this function:

• The partial derivative with respect to \(x\) is \(\frac{\partial f}{\partial x} = -2x\).
• The partial derivative with respect to \(y\) is \(\frac{\partial f}{\partial y} = 1\).

These values are combined to form the gradient vector \(abla f = (-2x, 1)\). This vector shows both the direction and the rate of change of the function. At the point \((2, 5)\), the gradient becomes \((-4, 1)\). This means the function increases most quickly in that direction.

Partial Derivatives

Partial derivatives are like taking a regular derivative, but for functions of multiple variables. Instead of looking at how a function changes overall, partial derivatives show how it changes with respect to one variable, holding others constant. This is extremely useful when dealing with functions of more than one variable, like \(f(x, y) = y - x^2\).

The partial derivative with respect to \(x\) for our function is \(-2x\). This signifies how much the function value changes when \(x\) is varied, keeping \(y\) constant. Similarly, the partial derivative with respect to \(y\) is \(1\), indicating a linear change at the same rate regardless of the value of \(y\).

Understanding partial derivatives is key to grasping how different variables impact the function separately. This concept is critical not just in sketching level curves, but also in fields like optimization and economics.

Parabola

A parabola is a symmetrical, curved shape you might be familiar with from quadratic equations. Its simplest form is \(y = x^2\). In our case, the level curve given by \(y = x^2 + 1\) shifts this basic parabola upwards by one unit.

• The vertex of this parabola is at \((0, 1)\), indicating the lowest point of the curve when facing upwards.
• It is symmetric about the y-axis, meaning if you fold the plane along the y-axis, both sides of the parabola would match perfectly.

Parabolas are part of the family of conic sections and appear in various forms across mathematics and physics. For example, they describe trajectories and beams of light or sound waves. When sketching a parabola like this, it's important to note its vertex and the direction it opens. For our equation, it easily passes through the point \((2, 5)\) as demonstrated in the solution.