Calculus 1 : Graphing Functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #3 : How To Graph Functions Of Curves

A1         B2

C3         D4

For which of the graphs of  above is the following statment true?

Possible Answers:

A

D

B

C

Correct answer:

A

Explanation:

The condition reads 'the limit of f of x as x approaches three from the left'.  Note that we're not looking at x as it approached negative three.  Only A and B have infinite limites at positive 3 (C and D show limits as x approaches negative 3), so the answer must be one of these.  We can see that from the left, B approaches positive infinity at x=3 from the left, while A approaches negative infinity at x=3 from the left, so the correct answer is A.

Example Question #4 : How To Graph Functions Of Curves

On what interval is the following function decreasing?

Possible Answers:

Correct answer:

Explanation:

Since this is a quadratic function with a positive first coefficient, we know it only has one local minimum and no other extrema. So all we have to do is find the x coordinate corresponding to this value and we will have one endpoint of our interval, and it will be decreasing constantly as you approach this point from the left.

Recall, 

So, if we take the derivative of this quadratic, we get

So this is where our interval ends. Since there are no other local extrema, we know our interval is 

.

Example Question #1 : How To Graph Functions Of Curves

Screen shot 2015 07 10 at 8.27.12 pm

Which of the following is true about the twice-differentiable function  above?

Possible Answers:

Correct answer:

Explanation:

Since the function  is increasing at , and since  is below the x-axis, .

Furthermore, there exists an inflection point at , where the concavity of function changes.

Thus at .

Therefore, the correct answer is 

Example Question #6 : How To Graph Functions Of Curves

Which one of the following could be the integral of ?

Possible Answers:

Correct answer:

Explanation:

Since the functions are added together, we can take each one seperately and add the results together.

The integral of  is  since you must apply for the chain rule of the .

The integral of  will be using the power rule

,

which means it will equal 

, which turns into ,

so combining these gives 

 

as the integral, making  the only equation that satisfies this quality. 

Example Question #7 : How To Graph Functions Of Curves

Exponential Function

What is the graph of the folloiwng function:

Possible Answers:

3

1

2

4

None of the above

Correct answer:

1

Explanation:

Use the following values to plot the graph:

, , ,

Example Question #8 : How To Graph Functions Of Curves

Trigonometric Function

Graph the folloiwng function:

Possible Answers:

5

None of the above

1

2

4

Correct answer:

5

Explanation:

Plot the graph for the following values:

Example Question #1 : How To Graph Functions Of Area

Screen shot 2015 07 10 at 12

Graph of a piecewise-linear function , for , is shown above.

Find

.

Possible Answers:

Correct answer:

Explanation:

Find the area under the graph  from . To do this break the graph up into triangles, squares, and rectangles to calculate the individual areas over smaller intervals and then add them all together.

The areas are added to be:

Therefore,

Example Question #2 : How To Graph Functions Of Area

Find the area bounded by the curve and the -axis over the interval .

Possible Answers:

Correct answer:

Explanation:

The curve is in quadrant one over the given interval, which gives us the bounds of integration.  Evaluating this definite integral yields the area we are after.

In order to performe the antiderivative, let .  It follows that .  Therefore

so

Example Question #1 : How To Graph Functions Of Area

Find the area bounded by the curve and the -axis over the interval .

Possible Answers:

Correct answer:

Explanation:

The curve is positive over the given interval, so the endpoints of the interval will mark the bounds of integration.  This function is very easy to integrate because the derivative of is itself!

Example Question #4 : How To Graph Functions Of Area

Find the area bounded by the curve and the -axis over the interval .

Possible Answers:

Correct answer:

Explanation:

This function is positve over the given interval, so the endpoints of the interval mark the bounds of integration.  It is a straightforward integration that is solvable with u-substitution.  Let so .  This means

so

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