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 does the slope of the tangent line on the graph of an implicit function represent?
It represents the value of $\frac{dy}{dx}$ at that point, indicating the rate of change of y with respect to x.
How can you identify critical points on the graph of an implicit function?
Look for points where the tangent line is horizontal (local max/min) or vertical (undefined derivative).
What does a concave up section of the graph of an implicit function indicate?
The second derivative is positive in that region.
What does a concave down section of the graph of an implicit function indicate?
The second derivative is negative in that region.
How can you identify points of inflection on the graph of an implicit function?
Look for points where the concavity changes (from concave up to concave down or vice versa).
What does a vertical tangent line on the graph of an implicit function indicate?
The derivative $\frac{dy}{dx}$ is undefined at that point.
How does the graph of an implicit function differ from an explicit function?
Implicit functions may not pass the vertical line test, and their graphs can be more complex.
How to interpret the graph of $x^2 + y^2 = 25$?
Circle with radius 5 centered at the origin. Top half has positive y values, bottom half has negative y values.
How to interpret the graph of $\frac{dy}{dx}$ of an implicit function?
Positive values indicate increasing function, negative values indicate decreasing function, zero values indicate critical points.
How to interpret the graph of $\frac{d^2y}{dx^2}$ of an implicit function?
Positive values indicate concave up, negative values indicate concave down, zero values indicate potential inflection points.
Explain how to find critical points of an implicit function.
Find $\frac{dy}{dx}$ using implicit differentiation, set it equal to 0 and undefined, and solve for x and y.
How does the sign of $\frac{dy}{dx}$ relate to the function's behavior?
Positive $\frac{dy}{dx}$ means the function is increasing; negative means decreasing.
How does the sign of $\frac{d^2y}{dx^2}$ relate to the function's concavity?
Positive $\frac{d^2y}{dx^2}$ means the function is concave up; negative means concave down.
Explain how to use the first derivative test to find local extrema.
Analyze the sign change of $\frac{dy}{dx}$ around a critical point. Positive to negative indicates a local maximum, negative to positive indicates a local minimum.
Explain how to use the second derivative test to determine concavity.
If $f''(x) > 0$, the function is concave up. If $f''(x) < 0$, the function is concave down.
What is the significance of a point of inflection?
It marks a change in the concavity of the function, where the second derivative changes sign.
How do you determine where an implicit function is increasing or decreasing?
Find $\frac{dy}{dx}$ and determine the intervals where it is positive (increasing) or negative (decreasing).
How do you determine the concavity of an implicit function?
Find $\frac{d^2y}{dx^2}$ and determine the intervals where it is positive (concave up) or negative (concave down).
Explain the chain rule in the context of related rates.
It relates the rates of change of different variables with respect to time, allowing us to find $\frac{dy}{dt}$ given $\frac{dx}{dt}$ and $\frac{dy}{dx}$.
What is the importance of drawing a diagram in related rates problems?
It helps visualize the relationships between variables and identify the equation that relates them.
