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.

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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.

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 $x^2 + y^2 = 16$ ?

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 $x^2 + y^2 = 16$ ?

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 $x^2 + y^2 = 25$ ?

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 $x^2 + y^2 = 9$ ?

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?

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?

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?

Area formula: $A = \pi r^2$ . 2. Differentiate with respect to time: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ .

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?

$2x + 2y \frac{dy}{dx} = 0$

All Flashcards

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.

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 $x^2 + y^2 = 16$ ?

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 $x^2 + y^2 = 16$ ?

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 $x^2 + y^2 = 25$ ?

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 $x^2 + y^2 = 9$ ?

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?

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?

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?

Area formula: $A = \pi r^2$ . 2. Differentiate with respect to time: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ .

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?

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?

$2x + 2y \frac{dy}{dx} = 0$