Problem 81 - Calculus Early Transcendentals

Question

Use the definition of the gradient (in two or three dimensions), assume that \(f\) and \(g\) are differentiable functions on \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3},\) and let \(c\) be a constant. Prove the following gradient rules. a. Constants Rule: \(\nabla(c f)=c \nabla f\) b. Sum Rule: \(\nabla(f+g)=\nabla f+\nabla g\) c. Product Rule: \(\nabla(f g)=(\nabla f) g+f \nabla g\) d. Quotient Rule: \(\nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}}\) e. Chain Rule: \(\nabla(f \circ g)=f^{\prime}(g) \nabla g,\) where \(f\) is a function of one variable

Step-by-Step Solution

Answer

Question: Verify the gradient rules for differentiable functions f(x,y) = x^2 and g(x,y) = y^2, and explain the significance of these rules in calculus. Answer: We have verified the gradient rules for the given functions, f(x,y) = x^2 and g(x,y) = y^2, and found that the Constants Rule, Sum Rule, Product Rule, Quotient Rule, and Chain Rule hold true for these differentiable functions. These rules are significant in calculus because they allow us to manipulate and analyze combinations of differentiable functions, enabling us to solve real-world problems that involve rates of change and optimization.

Step 1: Multiply function f by a constant c

First, we will multiply f by a constant, c = 3. So the function becomes: cf = 3x^2.

Step 2: Find the gradient of cf and cf

Now, let's find the gradient of f and cf: ∇f = (\<∂f/∂x, ∂f/∂y>) = \<2x, 0> ∇(cf) = \<∂(cf)/∂x, ∂(cf)/∂y>) = \<6x, 0>

Step 3: Verify Constants Rule

Since ∇(cf) = c ∇f, we can say the constants rule has been verified. b. Sum Rule:

Step 1: Find the gradient of functions f and g separately

∇f = \<∂f/∂x, ∂f/∂y>) = \<2x, 0> ∇g = (\<∂g/∂x, ∂g/∂y>) = \<0, 2y>

Step 2: Calculate the sum of the gradients and the gradient of the sum

Now, let's add the gradients and find the gradient of the sum (f+g): Sum of Gradients: (∇f) + (∇g) = \<2x, 0> + \<0, 2y> = \<2x, 2y> Gradient of Sum: ∇(f+g) = \<∂(f+g)/∂x, ∂(f+g)/∂y> = \<2x, 2y>

Step 3: Verify Sum Rule

Since the gradient of the sum is equal to the sum of gradients, the sum rule is verified. c. Product Rule:

Step 1: Calculate the product of functions f and g

Product of f and g: fg = x^2 * y^2

Step 2: Calculate the gradient of the product and apply the Product Rule

Now, let's find the gradient of the product and apply the product rule (∇(fg) = (∇f)g + f(∇g)): Gradient of Product: ∇(fg) = \<∂(fg)/∂x, ∂(fg)/∂y> = \<2xy^2, 2x^2y> Product Rule: (∇f)g + f(∇g) = (\<2x, 0>)y^2 + x^2(\<0, 2y>) = \<2xy^2, 2x^2y>

Step 3: Verify Product Rule

As the gradient of the product equals to applying the product rule, the rule is verified. d. Quotient Rule:

Step 1: Calculate the quotient of the functions f and g

Quotient of f and g: f/g = x^2 / y^2

Step 2: Calculate the gradient of the quotient and apply the Quotient Rule

Now, let's find the gradient of the quotient and apply the quotient rule (∇(f/g) = (g(∇f) - f(∇g))/g^2): Gradient of Quotient: ∇(f/g) = \<∂(f/g)/∂x, ∂(f/g)/∂y> = \<2x/y^2,-2x^2/y^3> Quotient Rule: (g(∇f) - f(∇g))/g^2 = (\<2x, 0>)(y^2) - (x^2)(\<0, 2y>))/(y^2)^2 = (\<2x/y^2,-2x^2/y^3>

Step 3: Verify Quotient Rule

As the gradient of the quotient equals applying the quotient rule, the rule is verified. e. Chain Rule: Let's consider new function h(t) = t^3 and f(h) = h^2

Step 1: Calculate the derivative of f with respect to h, and the gradient of h

Derivative of f with respect to h: df/dh = 2h Gradient of h: ∇h = \<∂h/∂x, ∂h/∂y> = \<3x^2, 3y^2>

Step 2: Calculate the gradient of the composition (f ∘ h) and apply the Chain Rule

Now, let's find the gradient of the composition and apply the chain rule(∇(f ∘ h) = (f'(h)) ∇h): Gradient of Composition: ∇(f ∘ h) = \<6x^3, 6y^3> Chain Rule: (f'(h))∇h = (2h)\<3x^2, 3y^2> = \<6x^3, 6y^3>

Step 3: Verify Chain Rule

As the gradient of the composition equals applying the chain rule, the rule is verified.

Key Concepts

Gradient

The gradient is a vector of partial derivatives and provides a directional derivative, indicating the direction of steepest ascent for a function of several variables. For a function \( f(x, y) \), the gradient is written as \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \).
This vector points in the direction in which \( f \) increases most rapidly and its magnitude gives the rate of increase in that direction.
In physical terms, the gradient can be thought of as a multi-dimensional slope.

• The components of the gradient vector are the rates of change of the function with respect to each variable.
• The gradient is key in optimization problems, helping to find the maximum or minimum values by guiding the direction of steepest ascent or descent.

Understanding gradients is essential for fields like machine learning, where they are used to adjust weights via gradient descent to minimize error.

Differentiable Functions

A function is differentiable if it has a derivative at each point in its domain. This is crucial because differentiability ensures that the function is smooth and that small changes in input result in small changes in output.
Differentiable functions are the backbone of calculus and are pivotal in solving more complex mathematical models.

• For functions of several variables, differentiability implies that the partial derivatives exist and the gradient is defined.
• Differentiability allows the use of approximation techniques such as Taylor series, which is vital in numerical methods.

Recognizing a function as differentiable allows us to apply various calculus techniques, including gradient-related rules, to solve real-world problems more effectively.

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. In the context of partial derivatives, it's applied when a variable depends on one or more other variables, each of which then impacts the function.
Mathematically, for composite functions \( f(g(x)) \), the chain rule is expressed as \( abla(f \circ g) = f'(g) abla g \). This helps in finding the gradient of the composite function efficiently.

• The chain rule is especially useful when dealing with nested functions or when functions are expressed in parametric form.
• It breaks down complex differentiation problems into manageable steps by considering the rate of change at each stage.

By understanding the chain rule, you can handle differentiation of cascaded functions with ease, ensuring precise solutions in multi-dimensional calculus contexts.

Product Rule

The product rule allows us to find the derivative of the product of two functions. For functions of several variables, where \( f \) and \( g \) are differentiable, the product rule states: \( abla(fg) = (abla f)g + f(abla g) \).
This rule highlights how the gradient is distributed over the multiplication of two functions, ensuring each function's rate of change is accurately captured.

• The product rule is essential when functions multiply in scenarios like physics or engineering, representing phenomena such as force or energy.
• It helps maintain accuracy when differentiating expressions, particularly when dealing with polynomials or power series.

Understanding and applying the product rule allows you to dissect and differentiate multiplied expressions correctly, which is critical in precise mathematical modeling and calculation.