What is an implicit function?

A function defined by an equation with multiple variables on the same side, not explicitly solved for one variable.

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What is an implicit function?

A function defined by an equation with multiple variables on the same side, not explicitly solved for one variable.

What is implicit differentiation?

A technique to find the derivative of an implicit function by differentiating both sides of the equation with respect to a variable.

Define critical points in the context of implicit functions.

Points where the derivative $\frac{dy}{dx}$ is either 0 or undefined, indicating potential minima or maxima.

What is a point of inflection?

A point where the concavity of a function changes, and the second derivative $f''(x) = 0$ or is undefined.

What does $\frac{dy}{dx} = 0$ indicate?

Potential critical points where the tangent line is horizontal, possibly indicating a local minimum or maximum.

What does an undefined $\frac{dy}{dx}$ indicate?

Potential critical points where the tangent line is vertical, possibly indicating a cusp or vertical tangent.

What is the first derivative test?

A method to determine if a critical point is a local minimum or maximum by analyzing the sign change of the first derivative around that point.

What is the second derivative test?

A method to determine the concavity of a function and identify points of inflection using the sign of the second derivative.

What does a positive second derivative indicate?

The function is concave up.

What does a negative second derivative indicate?

The function is concave down.

What is the notation for the derivative of y with respect to x?

$\frac{dy}{dx}$

What is the general form of an implicit function?

$F(x, y) = 0$

Chain rule formula to find $\frac{dy}{dt}$ ?

$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$

Pythagorean theorem formula?

$a^2 + b^2 = c^2$

How to denote the second derivative of a function?

$f''(x)$ or $\frac{d^2y}{dx^2}$

Formula to find critical points?

$\frac{dy}{dx} = 0$ or $\frac{dy}{dx}$ is undefined

How to express implicit differentiation?

$\frac{d}{dx} [F(x, y)] = 0$

Formula for the first derivative test?

If $f'(x)$ changes from positive to negative at $x=k$ , then $f(x)$ has a relative maximum at $x=k$ . If $f'(x)$ changes from negative to positive at $x=k$ , then $f(x)$ has a relative minimum at $x=k$ .

Formula for the second derivative test?

If $f''(x) > 0$ , then $f(x)$ is concave up. If $f''(x) < 0$ , then $f(x)$ is concave down.

How to express the derivative of $x^2 + y^2 = 25$ with respect to x?

$2x + 2y \frac{dy}{dx} = 0$

How to find $\frac{dy}{dx}$ for $x^2 + y^2 = 4$ ?

Differentiate both sides: $2x + 2y\frac{dy}{dx} = 0$ . Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = -\frac{x}{y}$ .

Steps to find critical points for $x^2 + y^2 = 16$ ?

Find $\frac{dy}{dx}$ . 2. Set $\frac{dy}{dx} = 0$ and undefined. 3. Solve for x and y.

How to determine if $(0, 4)$ is a local max/min for $x^2 + y^2 = 16$ ?

Find $\frac{dy}{dx}$ . 2. Evaluate $\frac{dy}{dx}$ around $(0, 4)$ . 3. Check for sign change.

How to find $\frac{dy}{dt}$ given $\frac{dx}{dt} = 3$ for $x^2 + y^2 = 25$ ?

Find $\frac{dy}{dx}$ . 2. Use chain rule: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$ . 3. Substitute values.

How to find the concavity of $x^2 + y^2 = 9$ ?

Find $\frac{dy}{dx}$ . 2. Find $\frac{d^2y}{dx^2}$ . 3. Determine intervals where $\frac{d^2y}{dx^2}$ is positive or negative.

How to find points of inflection for an implicit function?

Find $f''(x)$ . 2. Set $f''(x) = 0$ and solve for $x$ . 3. Check for concavity change around these points.

Steps to solve a related rates problem?

Identify variables and rates. 2. Find equation relating variables. 3. Differentiate with respect to time. 4. Substitute and solve.

How to check if a critical point is a local max or min?

Use the first derivative test (sign change of $f'(x)$ ) or the second derivative test (sign of $f''(x)$ ).

How to determine the equation relating variables in a related rates problem involving a right triangle?

Use the Pythagorean theorem: $a^2 + b^2 = c^2$ .

How to find the rate at which the area of a circle is changing?

Area formula: $A = \pi r^2$ . 2. Differentiate with respect to time: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ .

All Flashcards

What is an implicit function?

A function defined by an equation with multiple variables on the same side, not explicitly solved for one variable.

What is implicit differentiation?

A technique to find the derivative of an implicit function by differentiating both sides of the equation with respect to a variable.

Define critical points in the context of implicit functions.

Points where the derivative $\frac{dy}{dx}$ is either 0 or undefined, indicating potential minima or maxima.

What is a point of inflection?

A point where the concavity of a function changes, and the second derivative $f''(x) = 0$ or is undefined.

What does $\frac{dy}{dx} = 0$ indicate?

Potential critical points where the tangent line is horizontal, possibly indicating a local minimum or maximum.

What does an undefined $\frac{dy}{dx}$ indicate?

Potential critical points where the tangent line is vertical, possibly indicating a cusp or vertical tangent.

What is the first derivative test?

A method to determine if a critical point is a local minimum or maximum by analyzing the sign change of the first derivative around that point.

What is the second derivative test?

A method to determine the concavity of a function and identify points of inflection using the sign of the second derivative.

What does a positive second derivative indicate?

The function is concave up.

What does a negative second derivative indicate?

The function is concave down.

What is the notation for the derivative of y with respect to x?

$\frac{dy}{dx}$

What is the general form of an implicit function?

$F(x, y) = 0$

Chain rule formula to find $\frac{dy}{dt}$ ?

$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$

Pythagorean theorem formula?

$a^2 + b^2 = c^2$

How to denote the second derivative of a function?

$f''(x)$ or $\frac{d^2y}{dx^2}$

Formula to find critical points?

$\frac{dy}{dx} = 0$ or $\frac{dy}{dx}$ is undefined

How to express implicit differentiation?

$\frac{d}{dx} [F(x, y)] = 0$

Formula for the first derivative test?

Formula for the second derivative test?

If $f''(x) > 0$ , then $f(x)$ is concave up. If $f''(x) < 0$ , then $f(x)$ is concave down.

How to express the derivative of $x^2 + y^2 = 25$ with respect to x?

$2x + 2y \frac{dy}{dx} = 0$

How to find $\frac{dy}{dx}$ for $x^2 + y^2 = 4$ ?

Differentiate both sides: $2x + 2y\frac{dy}{dx} = 0$ . Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = -\frac{x}{y}$ .

Steps to find critical points for $x^2 + y^2 = 16$ ?

Find $\frac{dy}{dx}$ . 2. Set $\frac{dy}{dx} = 0$ and undefined. 3. Solve for x and y.

How to determine if $(0, 4)$ is a local max/min for $x^2 + y^2 = 16$ ?

Find $\frac{dy}{dx}$ . 2. Evaluate $\frac{dy}{dx}$ around $(0, 4)$ . 3. Check for sign change.

How to find $\frac{dy}{dt}$ given $\frac{dx}{dt} = 3$ for $x^2 + y^2 = 25$ ?

Find $\frac{dy}{dx}$ . 2. Use chain rule: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$ . 3. Substitute values.

How to find the concavity of $x^2 + y^2 = 9$ ?

Find $\frac{dy}{dx}$ . 2. Find $\frac{d^2y}{dx^2}$ . 3. Determine intervals where $\frac{d^2y}{dx^2}$ is positive or negative.

How to find points of inflection for an implicit function?

Find $f''(x)$ . 2. Set $f''(x) = 0$ and solve for $x$ . 3. Check for concavity change around these points.

Steps to solve a related rates problem?

Identify variables and rates. 2. Find equation relating variables. 3. Differentiate with respect to time. 4. Substitute and solve.

How to check if a critical point is a local max or min?

Use the first derivative test (sign change of $f'(x)$ ) or the second derivative test (sign of $f''(x)$ ).

How to determine the equation relating variables in a related rates problem involving a right triangle?

Use the Pythagorean theorem: $a^2 + b^2 = c^2$ .

How to find the rate at which the area of a circle is changing?

Area formula: $A = \pi r^2$ . 2. Differentiate with respect to time: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ .