What is the formula for the equation of a circle centered at the origin?
x2+y2=r2
Flip to see [answer/question]
Flip to see [answer/question]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Flip
Revise later
SpaceTo flip
If confident
All Flashcards
What is the formula for the equation of a circle centered at the origin?
x2+y2=r2
How to find dxdy using implicit differentiation?
Differentiate both sides of the equation with respect to x, remembering the chain rule for terms involving y. Then, solve for dxdy.
What is the general form of an ellipse centered at the origin?
a2x2+b2y2=1
If x2+y2=r2, what is dxdy?
dxdy=−yx
If x2+4y2=16, what is y in terms of x?
y=±2116−x2
What is the formula for the slope of a tangent line?
m=dxdy
What is the general formula for implicit differentiation?
dxdf(x,y)=∂x∂f+∂y∂fdxdy
What is the formula for the derivative of yn with respect to x?
dxd(yn)=nyn−1dxdy
What is the equation for the tangent line at a point (x0,y0)?
y−y0=dxdy∣(x0,y0)(x−x0)
What is the formula to solve for y in the equation x2+y2=1?
y=±1−x2
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 does a positive slope on the graph of an implicit function indicate?
As x increases, y also increases. The function is increasing at that point.
What does a negative slope on the graph of an implicit function indicate?
As x increases, y decreases. The function is decreasing at that point.
What does a horizontal tangent line on the graph of an implicit function indicate?
The derivative is zero, indicating a local maximum, local minimum, or a point of inflection.
What does a vertical tangent line on the graph of an implicit function indicate?
The derivative is undefined, often indicating a cusp or a sharp turn in the graph.
How can you visually identify the domain and range of an implicit function from its graph?
Domain: the set of all x-values covered by the graph. Range: the set of all y-values covered by the graph.
How does the symmetry of the graph relate to the implicit equation?
If replacing x with -x or y with -y leaves the equation unchanged, the graph has symmetry about the y-axis or x-axis, respectively.
What does the concavity of the graph tell us about the second derivative?
Concave up: second derivative is positive. Concave down: second derivative is negative.
How do you find the intervals where the function is increasing or decreasing from the graph?
Increasing: the graph has a positive slope. Decreasing: the graph has a negative slope.
What can you infer from the graph if the implicit function is periodic?
The function repeats its values in regular intervals. This can be confirmed by checking if f(x+T)=f(x) for some constant T.
How does the graph of x2+y2=r2 relate to its derivative?
The graph is a circle. The derivative gives the slope of the tangent line at any point on the circle. The slope will be undefined at x=±r, and zero at y=±r.