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Use the table of values to estimate:

The table represents x at 3.0001 and f(x) at 6.001, so simply follow the pattern.
Construct a table and find the indicated limit.
If h(x) = , then find the indicated limit.

 

Part 1
Plug the values in and solve.
Compute each separately.
For example:  √6.9 +(2) / 6.9 – 7 = -46.2679
 

 
 = -46.2679



 

 

 
The cost​ (in dollars) for manufacturing a particular DVD is C(x) = 7x + 16,000​, where x is the number of DVDs produced.
Recall from the previous chapter that the average cost per​ DVD, denoted by  is found by dividing C(x) by x.
Find and interpret

 

First, we plug in the values 7, and 16,000 by creating a fraction.


Now we plug in the value


= 0

So, we are left with 7
The cost​ (in dollars) for manufacturing a particular DVD is C(x) = 4x + 15,000​, where x is the number of DVDs produced.
Recall from the previous chapter that the average cost per​ DVD, denoted by  is found by dividing C(x) by x.
Find and interpret    

This problem is the same steps as above!

 
Use the properties of limits to help decide whether the limit exists. If the limit​ exists, find its value.

We first find the derivative of 
 
Then we plug in the value 6.
We can find the derivative of the numerator and denominator separately.
The formula for finding a derivative is n^{n – 1} with “n” being the exponent.
So, what we do is multiply the value of the exponent by the number in front of it and subtract 1 from the exponent.
For instance, if we have x² and need to find the derivative we multiply 2 (which is the exponent) by 1,
and subtract 1 from the exponent. So, it would be 2(1)^{2 – 1}.
The derivative of x² is 2x.
If a number value has no exponent, then the derivative of that number is 0.
For instance, the derivative of 1 = 0, 2 = 0, 3 = 0, and so on.

The derivative of x² – 36 = 2x
The derivative of x – 6 = 0
We are left with 2x. Now we plug in the value 6.
2(6) = 12


Use properties of limits to find the indicated limit.
It may be necessary to rewrite the expression before limit properties can be applied.

This is the same as above.
Find the derivative and plug in the value of 3.
The derivative of the limit is in fraction form: 2x + 4 / 2x
2x + 4 / 2x
We plug in the values.
2(3) + 4 / 2(3) =     10 / 6
Now we reduce to the fraction

Use the properties of limits to help decide whether the limit exists. If the limit​ exists, find its value.

This is the same as above.
Find the derivative and plug in the value of 6.
I can do this in my head because I have done so many like this.
So, you can practice this on paper.

Use the properties of limits to help decide whether the limit exists. If the limit​ exists, find its value.

 

This is similar, however, all we need to do is find the derivative of

 and simplify.
 
After finding the derivative, we are left with x / x
the answer is 1 because the x’s cancel out.
 

Use the properties of limits to help decide whether the limit exists. If the limit​ exists, find its value.

 

This is a little trick so I will break the problem down.

We need to find the derivative of     
First, we find the derivative of 9x2 + 2x = 18x + 2
Next, we find the derivative of 4x2 – 7x + 1 = 8x – 7
 
So, we are left with a fraction   
 
Since we will be plugging in the value  for x, 18 and 8 will become
discounted from the rest of the values in the fraction, and the right side will be cancelled out.
So, we will have
       Then we cancel the right side of the fraction
 and we are left with
So, we simplify the fraction to        

 

 

This is the same as above.
If you don’t understand any of this, please contact me. I will take the time to explain.


 

This is the same as above.
If you don’t understand any of this, please contact me. I will take the time to explain.

 
Find all values x = a where the function is discontinuous.
For each value of​ x, give the limit of the function as x approaches.
a.      Be sure to note when the limit​ doesn't exist.

 

Since this problem involves f(x), which is a function, and the value of the function is not defined,
(example x --> 3), then f(x) is discontinuous (or vanishes). In this case, it vanishes at 0, 2
Naturally, the limits do not exist for either value, as well, because the limit is not approaching any value and vanishes.
   
Find all values x = a where the function is discontinuous.
For each value of​ x, give the limit of the function as x approaches
a. Be sure to note when the limit​ doesn't exist.
F(x) = 2x² + 3x -3
 

 
Explanation
Solve 2x² + 3x = -3 for x
x = -1, and -1/2
Solve the limit for x = -4 for both.
When both limits = the same, then f is continuous
   
The cost to transport a mobile home depends on the​ distance, x, in miles that the home is moved.
Let​ C(x) represent the cost to move a mobile home x miles. One firm charges as follows.
 
 
A.     Falls in between 0 – 150 miles so the cost is $4 per mile, and we multiply 103 by 4.00 = 412
B.      Falls in between 0 – 150 miles so the cost is $4 per mile, and we multiply 150 by 4.00 = 600
       And so on… Suppose that f(t) = t² + 5t - 5
What is the average rate of change of f(t) over the interval 1 to 2
The average rate of change of f(t) over the interval 1 to 2 is ___

Answer: 8
 
f(b) – f(a) / b – a
f(t) = t² + 5t – 5
f(1) = (1)² + 5(1) – 5
f(1) = 1 + 5 – 5 = 1
f(2) = (2)² + 5(2) – 5
f(2) = 4 + 10 – 5 = 9
f(b) – f(a) / b – a
9 – 1/ 2 - 1 = 8
The average rate of change of f(t) over the interval 1 to 2 is 8
Check
https://www.emathhelp.net/calculators/calculus-1/average-rate-of-change-calculator/
   

Find the average rate of change of the function over the given interval.
y = √2x + 3   between x = 3 and x = 11
f(b) – f(a) / b – a
f(3) = √2(3) + 3  =  √6 + 3 = √9 = 3
f(3) = √2(11) + 3  =  √22 + 3 = √25 = 5
f(b) – f(a) / b – a
5 – 3 / 11 - 3 =
 
   
Find the average rate of change of the function over the given intervals
f(x) = 7x³ + 7;                   a) [2, 4],          b) [=3, 3]

 
 
We simply plug in the values, subtract (larger from smaller,
and divide by the larger value of the set minus the smaller value.
a)      7(4)³ + 7 – 7(2)³+ 7 / 4 – 2 =   455 – 63 / 2 =  392 / 2 =                                  196
 
The next one is tricky. Watch carefully how I factor the values.
b)     7(3)³ + 7 – 7(-3)³ + 7 / 3 – (-3) =          196 – (-182) / 6 =        378 / 6 =          63
   
Find the average rate of change for the function over the given interval.
y= e^x   between x = -4 and x = -2
 

 
x = -4 x = -2
e = 2.71828
a = 2.71828^-4 = .0183
b = 2.71828^-2 = .1353
f(b) – f(a) / b – a
.1353 - .0183 / -2 – (-4) = .1353 - .0183 / -2 + 4
.117 / 2 =         0.0585
 
Check
https://www.emathhelp.net/calculators/calculus-1/average-rate-of-change-calculator/  
Suppose the position of an object moving in a straight line is given by s(t) = 5t² + 3t + 5. Find the instantaneous velocity when t = 4.
What expression can be used to find the instantaneous velocity at the given​ time?
 
The instantaneous velocity at t = 4 is 43
 

 
Part 1: the limit definition of derivative is
 

And we are given the value of x, so we simply plug in the value of x and replace f with s because we are given the
The formula for the straight line at “s(t)” = 5t² + 3t + 5
 is the logical answer to part 1.
 
Part 2: is a lot easier. We already covered the steps to finding the derivative of a function.
So, we need to find the derivative of s(t) = 5t² + 3t + 5 and plug in the value of t = 4.
Remember the derivative of
5t² is 5(2)t^{2 – 1} = 10t
3t is 1(3)^{1 – 1} = 3
The derivative of a number without a variable or exponent is 0
So, we are left with 10t + 3
Now we plug in the value 4
10(4) + 3 = 40 + 3 = 43
  Decide from the graph whether each limit exists. If a limit exists, estimate it’s value.
What is the value of the limit?

Watch the graph carefully.
We are given x->-6 and the limit meets on the graph at positive 4
We are given x->-5 and the limit meets on the graph at positive 5 Use the graph to find the following limits.
Find

 

Watch the graph carefully.
We are given x-> -2 and the limit meets on the graph at +1
We are given x-> 0 and the limit does not meet. It vanishes, therefore the limit does not exist

Use the graph to find the following limits and function value for each value of a.

 

 

 
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Use the graph to find the following limits and function value for each of the value of a.

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Refer to the figure below to find the limit lim f(x).
What is the limit

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  Use the graph of f(x) = e^x below to complete a and b below

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  Use the graph of f(x) = ln x below to complete a and b below

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