### All Calculus 3 Resources

## Example Questions

### Example Question #1 : Gradient Vector, Tangent Planes, And Normal Lines

Find the equation of the tangent plane to at .

**Possible Answers:**

**Correct answer:**

First, we need to find the partial derivatives in respect to , and , and plug in .

,

,

,

Remember that the general equation for a tangent plane is as follows:

Now lets apply this to our problem

### Example Question #2 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Trigonometric derivative:

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at at the point

x:

y:

z:

### Example Question #3 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Trigonometric derivative:

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at at the point

x:

y:

z:

### Example Question #4 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at at the point

x:

y:

z:

### Example Question #1 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

It is essentially the slope of a multi-dimensional function at any given point.

Looking at at the point

x:

y:

z:

### Example Question #6 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

It is essentially the slope of a multi-dimensional function at any given point.

Looking at at the point

x:

y:

z:

### Example Question #7 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

It is essentially the slope of a multi-dimensional function at any given point.

Looking at at the point

x:

y:

z:

### Example Question #8 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

It is essentially the slope of a multi-dimensional function at any given point.

Looking at at the point

x:

y:

### Example Question #9 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

It is essentially the slope of a multi-dimensional function at any given point.

Looking at at the point

x:

y:

### Example Question #10 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function at the point

**Possible Answers:**

**Correct answer:**

To consider finding the slope, let's discuss the topic of the gradient.

It is essentially the slope of a multi-dimensional function at any given point.

Looking at at the point

x:

y: