All Calculus 3 Resources
Example Questions
Example Question #1 : Calculus 3
Suppose the vectors and are orthogonal. Find all real values of .
Example Question #2 : Calculus 3
The vector-valued function paremeterizes a curve , where .
Find a tangent vector to at the point .
Example Question #2 : Calculus 3
Let .
Find .
Example Question #4 : Calculus 3
Let .
Find its linear approximation at .
Hint: Use Taylor's formula.
Example Question #3 : Calculus 3
Let . Which of the following is equal to ?
Notice that can be expressed as a composite function, i.e. a function within a function. If we let and , then . In order to differentiate , we will need to apply the Chain Rule, as shown below:
First, we need to find , which equals .
Then, we need to find by applying the Product Rule.
The answer is .
Example Question #4 : Calculus 3
Evaluate:
cannot be determined
First, we can write out the first few terms of the sequence , where ranges from 1 to 3.
Notice that each term , is found by multiplying the previous term by . Therefore, this sequence is a geometric sequence with a common ratio of . We can find the sum of the terms in an infinite geometric sequence, provided that , where is the common ratio between the terms. Because in this problem, is indeed less than 1. Therefore, we can use the following formula to find the sum, , of an infinite geometric series.
The answer is .
Example Question #171 : Functions, Graphs, And Limits
Find if .
We will have to find the first derivative of with respect to using implicit differentiation. Then, we can find , which is the second derivative of with respect to .
We will apply the chain rule on the left side.
We now solve for the first derivative with respect to .
In order to get the second derivative, we will differentiate with respect to . This will require us to employ the Quotient Rule.
We will replace with .
But, from the original equation, . Also, if we solve for , we obtain .
The answer is .
Example Question #5 : Interpretations And Properties Of Definite Integrals
Which of the following represents the graph of the polar function in Cartestian coordinates?
First, mulitply both sides by r.
Then, use the identities and .
The answer is .
Example Question #6 : Calculus 3
We can use the substitution technique to evaluate this integral.
Let .
We will differentiate with respect to .
, which means that .
We can solve for in terms of , which gives us .
We will also need to change the bounds of the integral. When , , and when , .
We will now substitute in for the , and we will substitute for .
The answer is .
Example Question #4 : Fundamental Theorem Of Calculus
Evaluate the following limit:
Does not exist.
First, let's multiply the numerator and denominator of the fraction in the limit by .
As becomes increasingly large the and terms will tend to zero. This leaves us with the limit of .
.
The answer is .