Calculus 1 : Meaning of Functions
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Example Questions
Example Question #31 : Meaning Of Functions
Evaluate:
The limit does not exist.
In order to evaluate the limit, lets factor the numerator.
Now we can simplify this expression to
Now we plug in 10.
Example Question #1755 : Functions
Evaluate:
The limit does not exist.
If we plug in 
Since we get 
L'Hopitals rule is if we have one of the following cases
where a is any real number, then
After applying L'Hopitals rule, we get
Now if we plug in 
Example Question #1756 : Functions
Let 
The total increase in people in Tucson from 
The rate of change in the number of people in 
The average rate of change of people between 
The average number of people between 
The number of cats in people in 
The total increase in people in Tucson from
The integral of the rate of change of people gives the change in the number of people over the time interval. However, it does not tell you how many people are present because it contains no information on how many people were present initially. (Think that if you integrate 
Example Question #32 : How To Find The Meaning Of Functions
Find the critical point(s) of 
To find the critical point(s) of a function 
Given 
Since 
The critical point 
Example Question #33 : How To Find The Meaning Of Functions
Find the critical point(s) of 
To find the critical point(s) of a function
Given 
Thus the derivative becomes, 
Since 
The critical point 
Example Question #32 : Meaning Of Functions
Which of the following is not a function?
A function is defined when each value of x yields a single value of y (ordered pairs). The expression
is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.
Example Question #32 : Meaning Of Functions
Evaluate:
The limit does not exist
To evaluate the limit, we can factor the numerator
Now we simplify and evaluate.
Example Question #33 : Meaning Of Functions
Evaluate:
In order to evaluate the limit, we need to factor the numerator
Now we can simplify the expression to
Now we can plug in 1 to get
Example Question #34 : Meaning Of Functions
Evaluate the limit:
None of the other answers
By applying L'Hôpital's rule, we can find the limit by evaluating
The function is now written as
Plugging in 0 gives us
Example Question #33 : Meaning Of Functions
Which of the following statements is true regarding the behavior of functions with respect to its derivatives?
The second derivative tells you when the function changes from decreasing to increasing and vice versa.
The natural log 
If at a point, the derivative is positive and the second derivative in negative, the function is decreasing at an increasing rate.
An example of a point of inflection is where the function goes from increasing at a decreasing rate to increasing at an increasing rate.
An example of a point of inflection is where the function goes from increasing at a decreasing rate to increasing at an increasing rate.
The statement
"An example of a point of inflection is where the function goes from increasing at a decreasing rate to increasing at an increasing rate."
is true. Point of inflection is when the function changes from concave up to concave down and vice versa. In the example above, the function changes from concave down (slopes are decreasing) to concave up (slopes are increasing).
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