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  1. Pre-Calculus
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What is an implicitly defined function?
An equation where x and y are related, but y is not explicitly isolated.
What is an explicit function?
A function where y is isolated on one side of the equation, expressed directly in terms of x.
Define the slope of a curve at a point.
The rate of change of y with respect to x (dy/dx) at that point, representing the tangent line's steepness.
What is a horizontal interval on a graph?
An interval where the rate of change of x with respect to y is zero (dx/dy = 0), resulting in a slope of zero.
What is a vertical interval on a graph?
An interval where the rate of change of y with respect to x is undefined, resulting in an undefined slope.
What does 'solving for y' mean in the context of implicit functions?
Isolating y in terms of x, which may result in one or more explicit functions representing parts of the original implicit relation.
Define the domain of an implicitly defined function.
The set of all possible x-values for which the function is defined.
Define the range of an implicitly defined function.
The set of all possible y-values that the function can take.
What is implicit differentiation?
A method used to find the derivative of an implicitly defined function by differentiating both sides of the equation with respect to x, treating y as a function of x.
What is the relationship between the graph of an implicitly defined function and the equation?
The graph is the set of all ordered pairs (x, y) that satisfy the equation.
What is the formula for the equation of a circle centered at the origin?
$x^2 + y^2 = r^2$
How to find $\frac{dy}{dx}$ using implicit differentiation?
Differentiate both sides of the equation with respect to x, remembering the chain rule for terms involving y. Then, solve for $\frac{dy}{dx}$.
What is the general form of an ellipse centered at the origin?
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
If $x^2 + y^2 = r^2$, what is $\frac{dy}{dx}$?
$\frac{dy}{dx} = -\frac{x}{y}$
If $x^2 + 4y^2 = 16$, what is y in terms of x?
$y = \pm \frac{1}{2}\sqrt{16 - x^2}$
What is the formula for the slope of a tangent line?
$m = \frac{dy}{dx}$
What is the general formula for implicit differentiation?
$\frac{d}{dx}f(x,y) = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}$
What is the formula for the derivative of $y^n$ with respect to x?
$\frac{d}{dx}(y^n) = ny^{n-1} \frac{dy}{dx}$
What is the equation for the tangent line at a point $(x_0, y_0)$?
$y - y_0 = \frac{dy}{dx}|_{(x_0, y_0)} (x - x_0)$
What is the formula to solve for y in the equation $x^2 + y^2 = 1$?
$y = \pm \sqrt{1-x^2}$
How to find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$?
1. Differentiate: $2x + 2y\frac{dy}{dx} = 0$. 2. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x}{y}$.
Steps to find the tangent line to $x^2 + y^2 = 1$ at $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$?
1. Find $\frac{dy}{dx} = -\frac{x}{y}$. 2. Evaluate at the point: $\frac{dy}{dx} = -1$. 3. Use point-slope form: $y - \frac{\sqrt{2}}{2} = -1(x - \frac{\sqrt{2}}{2})$.
How do you determine if an equation implicitly defines y as a function of x?
Try to solve for y. If you get a single expression for y in terms of x, then y is a function of x. If you get multiple expressions, it may not be.
How to find the domain and range of $x^2 + y^2 = 4$?
1. Solve for y: $y = \pm \sqrt{4 - x^2}$. 2. Domain: $-2 \le x \le 2$. 3. Range: $-2 \le y \le 2$.
How to approach related rates problems involving implicit functions?
1. Identify variables and rates. 2. Write the equation relating the variables. 3. Differentiate implicitly with respect to time. 4. Substitute known values and solve for the unknown rate.
How do you find the points where the tangent line is horizontal for $x^2 + y^2 = 1$?
1. Find $\frac{dy}{dx} = -\frac{x}{y}$. 2. Set $\frac{dy}{dx} = 0$, which implies $x = 0$. 3. Solve for y: $y = \pm 1$.
How do you find the points where the tangent line is vertical for $x^2 + y^2 = 1$?
1. Find $\frac{dy}{dx} = -\frac{x}{y}$. 2. Set the denominator to zero, i.e., $y = 0$. 3. Solve for x: $x = \pm 1$.
How to use implicit differentiation to find the second derivative $\frac{d^2y}{dx^2}$?
1. Find $\frac{dy}{dx}$ using implicit differentiation. 2. Differentiate $\frac{dy}{dx}$ with respect to x, again using implicit differentiation and the quotient rule if necessary. 3. Substitute the expression for $\frac{dy}{dx}$ to simplify.
How to find the equation of the normal line to an implicitly defined curve at a point?
1. Find $\frac{dy}{dx}$ using implicit differentiation. 2. Evaluate $\frac{dy}{dx}$ at the given point to find the slope of the tangent line. 3. The slope of the normal line is the negative reciprocal of the tangent line's slope. 4. Use the point-slope form of a line to find the equation of the normal line.
How do you find the slope of the tangent line to the curve $x^3 + y^3 = 6xy$ at the point (3,3)?
1. Differentiate implicitly: $3x^2 + 3y^2\frac{dy}{dx} = 6y + 6x\frac{dy}{dx}$. 2. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{2y-x^2}{y^2-2x}$. 3. Evaluate at (3,3): $\frac{dy}{dx} = \frac{6-9}{9-6} = -1$.