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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 #1 : 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 #2 : 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 #2 : Calculus 3
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 #3 : Calculus 3
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 #332 : Ap Calculus Ab
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 #3 : Calculus 3
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 .
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