All Calculus 3 Resources
Example Questions
Example Question #1 :Calculus 3
Suppose the vectorsandare orthogonal. Find all real values of.
Example Question #2 :Calculus 3
The vector-valued functionparemeterizes a curve, where.
Find a tangent vector toat 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可以表示为一个复合函数,即function within a function. If we letand, 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 findby applying the Product Rule.
The answer is.
Example Question #1 :Numerical Approximations To Definite Integrals
Evaluate:
cannot be determined
First, we can write out the first few terms of the sequence, whereranges 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, whereis the common ratio between the terms. Becausein 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 #3 :Calculus 3
Findif.
We will have to find the first derivative ofwith respect tousing implicit differentiation. Then, we can find, which is the second derivative ofwith 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 differentiatewith respect to. This will require us to employ the Quotient Rule.
We will replacewith.
But, from the original equation,. Also, if we solve for, we obtain.
The answer is.
Example Question #1 :Calculus 3
Which of the following represents the graph of the polar functionin Cartestian coordinates?
First, mulitply both sides by r.
Then, use the identitiesand.
The answer is.
Example Question #1 :Calculus 3
We can use the substitution technique to evaluate this integral.
Let.
We will differentiatewith respect to.
, which means that.
We can solve forin terms of, which gives us.
We will also need to change the bounds of the integral. When,, and when,.
We will now substitutein for the, and we will substitutefor.
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.
Asbecomes increasingly large theandterms will tend to zero. This leaves us with the limit of.
.
The answer is.
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