calc III Exam II Study

How do you find the velocity vector?

Derivative w/ respect to time of the position vector.

How do you find the acceleration vector?

Derivative w/ respect to time of the velocity vector.

How do you find speed?

Magnitude of the velocity vector.

If you have initial conditions with the velocity and position vector, you:

Integral of before + C = Initial Condition Vector <Xo, Yo> or <Uo, Vo>

How to find the length of the curve?

L = Integral from a to b of the sqrt of f’(t)² + g’(t)²+ h’(t)²

When can you use an arc length as a parameter

Whenever the curve is smooth and the magnitude of the velocity vector equals 1 everywhere

What is the format of the domain?

{f(x,y): …}

How do you format the range?

{z:….}

The domain of the a square root function is:

X² + Y² greater than or equal to 0

The domain of a fraction

Anywhere that does not equal 0 in the denominator.

What do level curves mean?

Like a 3D surface z = f(x, y) horizontally with a flat plane at a fixed height z = c

When does the limit not exist?

If two or more different paths produce two or more different limits.

What is the two path test?

For a limit of a function f(x, y) as (x, y) approaches a point (x_0, y_0) to exist, the function must approach the exact same value no matter what path you take to get there.

What does it mean for a function to be continuous?

Has no breaks, holes, jumps, or vertical asymptotes

When does the limit no exist?

When the function fails to approach a single, unique finite value.

What is differentiability?

The technical condition that ensures the function can be approximated by a tangent plane.

Does differentiability imply continuity?

Yes. If a function is differentiable at a point, it is guaranteed to be continuous at that point.

Does continuity imply differentiability?

No, A function can be continuous (having no gaps or jumps) yet still fail to be differentiable.

What do partial derivatives show?

Two or more rates of change.

What is Clairut’s Theorum?

Suppose f is defined on an open set that contains (a,b). If Fxy and Fyx are continuous on D, then Fxy(a,b) = Fyx(a,b).

how do you take a partial derivative?

When computing the partial derivative, treat the other variables as constants.

What are higher order derivatives?

Fxx, Fxy, Fyy, Fyx

What is locally linear?

The graph of a differentiable function looks like a straight line when you zoom in closely enough.

What is the gradient?

A vector that collects all the partial derivatives of a multivariable scalar function.

Equation for the gradient

What do you find the maximum rate of change ?

The magnitude of the gradient vector.

What is the gradient orthogonal to?

Level Curve

What is a directional derivative?

dot product of the gradient and unit vector

Chain rule with partial derivatives.

A function increases more rapidly at a point p(xo,yo) in the direction of the gradient with the rate of the magnitude of the gradient.

True

How do you find the equation of the tangent plane?

Use the partial derivatives of the surface function at a specific point. Plug in the respective points and put it in the equation: Z - Zo = Fx(x - xo) + Fy(y - yo)