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 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 or is undefined.
What does indicate?
Potential critical points where the tangent line is horizontal, possibly indicating a local minimum or maximum.
What does an undefined 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?
What is the general form of an implicit function?
Chain rule formula to find ?
Pythagorean theorem formula?
How to denote the second derivative of a function?
or
Formula to find critical points?
or is undefined
How to express implicit differentiation?
Formula for the first derivative test?
If changes from positive to negative at , then has a relative maximum at . If changes from negative to positive at , then has a relative minimum at .
Formula for the second derivative test?
If , then is concave up. If , then is concave down.
How to express the derivative of with respect to x?
How to find for ?
Differentiate both sides: . Solve for : .
Steps to find critical points for ?
- Find . 2. Set and undefined. 3. Solve for x and y.
How to determine if is a local max/min for ?
- Find . 2. Evaluate around . 3. Check for sign change.
How to find given for ?
- Find . 2. Use chain rule: . 3. Substitute values.
How to find the concavity of ?
- Find . 2. Find . 3. Determine intervals where is positive or negative.
How to find points of inflection for an implicit function?
- Find . 2. Set and solve for . 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 ) or the second derivative test (sign of ).
How to determine the equation relating variables in a related rates problem involving a right triangle?
Use the Pythagorean theorem: .
How to find the rate at which the area of a circle is changing?
- Area formula: . 2. Differentiate with respect to time: .