Differentiation Techniques, Indeterminate Forms, Partial Differentiation, and Mathematical Modelling

Flashcards covering implicit differentiation, indeterminate forms, L'Hopital's Rule, partial differentiation, average and instantaneous rates of change, marginal cost, and related rates.

What is the explicit form of defining a function f(x)?

y = f(x)

What is implicit differentiation?

A method to solve for the derivative of a function defined implicitly.

List the steps for implicit differentiation.

1. Differentiate x-terms normally. 2. Differentiate y-terms with respect to y and place dy/dx next to each. 3. Apply product, quotient, and other differentiation rules to x-y terms.

What is an indeterminate form in the context of limits?

A limit is in indeterminate form if its existence cannot be determined by simply knowing the value of its part.

Name some indeterminate forms.

0/0, 0 ⋅ (±∞), ±∞/±∞, ∞ − ∞, 0^0, (±∞)^0, 1^(±∞)

State L’Hopital’s Rule.

If lim (x→a) f(x) / g(x) is an indeterminate form (0/0 or ±∞/±∞), then lim (x→a) f(x) / g(x) = lim (x→a) f'(x) / g'(x), provided f and g are differentiable on an open interval I, except possibly at a ∈ I.

Define n-ary Cartesian product of n sets.

The set of all n-tuples (a1, a2, a3, …, an) where ak ∈ Ak for all k = 1, 2, 3, …, n. Denoted as A1 × A2 × A3 × … × An.

What is a function of n variables

A function of n variables is a function from an n-ary Cartesian product A1 × A2 × A3 × … × An to a set B.

Define partial derivative.

The partial derivative of f with respect to xk is the function denoted by ∂f/∂xk, such that its function value at any point in the domain of f is given by ∂f/∂xk = lim (Δxk→0) [f(x1, x2, …, xk + Δxk, …, xn) - f(x1, x2, …, xn)] / Δx_k, if the limit exists.

What are the alternative notations for partial derivatives?

∂f/∂xk = Dkf = f{xk} = ∂{xk}f

What are second-order partial derivatives?

Partial derivatives of first-order partial derivatives

Define average rate of change.

The average rate of change of y per unit change in x is the value Δy/Δx = [f(x + Δx) - f(x)] / Δx.

Give examples of concepts modeled as the average rate of change.

Average cost, acceleration, and velocity

Define instantaneous rate of change.

lim (Δx→0) Δy/Δx = lim (Δx→0) [f(x + Δx) - f(x)] / Δx = f'(x)

Define instantaneous velocity and acceleration.

Instantaneous velocity is v(t) = d/dt f(t), and instantaneous acceleration is a(t) = d^2/dt^2 f(t).

What is the marginal cost function?

The derivative of the total cost function C(x) with respect to x, i.e., d/dx C(x).

List the steps in solving problems in related rates.

1. Identify the two related variables. 2. Identify the equation relating the two variables. 3. Identify the two related rates. 4. Identify the common variable in the two related rates. 5. Identify the known rate and the rate to be found. 6. Differentiate the equation relating the two variables on both sides by the common variable. 7. Evaluate the unknown rate based on the given value of the variable.

What is the explicit form of defining a function f(x)?

y = f(x)

What is implicit differentiation?

A method to solve for the derivative of a function defined implicitly.

List the steps for implicit differentiation.

2. Differentiate x-terms normally. 2. Differentiate y-terms with respect to y and place dy/dx next to each. 3. Apply product, quotient, and other differentiation rules to x-y terms.

What is an indeterminate form in the context of limits?

A limit is in indeterminate form if its existence cannot be determined by simply knowing the value of its part.

Name some indeterminate forms.

0/0, 0 ⋅ (±∞), ±∞/±∞, ∞ − ∞, 0^0, (±∞)^0, 1^(±∞)

State L’Hopital’s Rule.

If lim (x→a) f(x) / g(x) is an indeterminate form (0/0 or ±∞/±∞), then lim (x→a) f(x) / g(x) = lim (x→a) f'(x) / g'(x), provided f and g are differentiable on an open interval I, except possibly at a ∈ I.

Define n-ary Cartesian product of n sets.

The set of all n-tuples (a1, a2, a3, …, an) where ak ∈ Ak for all k = 1, 2, 3, …, n. Denoted as A1 × A2 × A3 × … × An.

What is a function of n variables

A function of n variables is a function from an n-ary Cartesian product A1 × A2 × A3 × … × An to a set B.

Define partial derivative.

The partial derivative of f with respect to xk is the function denoted by ∂f/∂xk, such that its function value at any point in the domain of f is given by ∂f/∂xk = lim (Δxk→0) [f(x1, x2, …, xk + Δxk, …, xn) - f(x1, x2, …, xn)] / Δx_k, if the limit exists.

What are the alternative notations for partial derivatives?

∂f/∂xk = Dkf = f{xk} = ∂{xk}f

What are second-order partial derivatives?

Partial derivatives of first-order partial derivatives

Define average rate of change.

The average rate of change of y per unit change in x is the value Δy/Δx = [f(x + Δx) - f(x)] / Δx.

Give examples of concepts modeled as the average rate of change.

Average cost, acceleration, and velocity

Define instantaneous rate of change.

lim (Δx→0) Δy/Δx = lim (Δx→0) [f(x + Δx) - f(x)] / Δx = f'(x)

Define instantaneous velocity and acceleration.

Instantaneous velocity is v(t) = d/dt f(t), and instantaneous acceleration is a(t) = d^2/dt^2 f(t).

What is the marginal cost function?

The derivative of the total cost function C(x) with respect to x, i.e., d/dx C(x).

2. Identify the two related variables. 2. Identify the equation relating the two variables. 3. Identify the two related rates. 4