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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 $x^2 + y^2 = 16$?
1. 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 $x^2 + y^2 = 16$?
1. 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 $x^2 + y^2 = 25$?
1. 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 $x^2 + y^2 = 9$?
1. 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?
1. 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?
1. 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?
1. Area formula: $A = \pi r^2$. 2. Differentiate with respect to time: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$.
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.