Calculus 1 : Curves

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #5 : How To Find Local Minimum Graphing Functions Of Curves

Given the equation of a graph is , find the coordinates of the local minimum(s) if any are present.

Possible Answers:

No local minimum

Correct answer:

Explanation:

To find the local minimum of any graph, you must first take the derivative of the graph equation, set it equal to zero and solve for .  To take the derivative of this equation, we must use the power rule,  

.  

We also must remember that the derivative of a constant is 0. Taking the derivative of the graph equation, we obtain the slope equation . Solving for x when the equation is set to 0, we obtain .  

In order for  to be a local minimum, the slope must increase as it passes 2 from the left.  Plugging in 1 and 3 into the slope equation, we find that the slope is in fact increase from -4 to 4, therefore  is a local minimum.  

Plugging  back into the original graph equation to solve for , we find the coordinates of the local minimum for this graph is in fact .

Example Question #6 : How To Find Local Minimum Graphing Functions Of Curves

Find any local maxima or minima of  on the interval .

Possible Answers:

Global Maximum at:

Global Minimum at:

Global Maximum at: 

Global Minimum at:

Correct answer:

Global Minimum at:

Explanation:

Find any local maxima or minima of f(x) on the interval [-10,10]

To begin finding local mins and maxes we need to take the first derivative of the above function.

Local minimum and maximums occur wherever the first derivative is 0. 

 

Find the y coordinate via:

So the first bit of our answer:

But is it a maximum or minimum?

To find that, we need to know if the function is concave up or concave down at the point.

To test concavity we need the second derivative:

The second derivative is positive everywhere, so this function is concave up everywhere, making this a local minimum.

Example Question #7 : How To Find Local Minimum Graphing Functions Of Curves

Find the local maximum of the function .

Possible Answers:

There are none.

Correct answer:

Explanation:

First, find the derivative of the function. 

.

Then we find that the points where the derivative equals 0 are at  and . Then picking points on the intervals between these critical points and plugging them into the derivative. Where the results are negative, the function is decreasing. Where they are positive, the function in increasing.

Using this method we see that in the on the interval  the function is increasing. On  the function is decreasing. On  the function is increasing. Because the function is increasing to the left of  and decreasing to the right,  is a local maximum. Plugging it into the function, we can find the y value.      

Example Question #8 : How To Find Local Minimum Graphing Functions Of Curves

Find the local minimum of the function .

Possible Answers:

There is none. 

Correct answer:

Explanation:

First, find the derivative of the function. 

.

Then we find that the points where the derivative equals 0 are at  and . Then picking points on the intervals between these critical points and plugging them into the derivative. Where the results are negative, the function is decreasing.

Where they are positive, the function in increasing. Using this method we see that in the on the interval  the function is increasing. On  the function is decreasing. On  the function is increasing. Because the function is decreasing to the left of  and increasing to the right,  is a local minimum. Plugging it into the function, we can find the y value.      

Example Question #9 : How To Find Local Minimum Graphing Functions Of Curves

What is the least -value for the function ?

Possible Answers:

Correct answer:

Explanation:

 To find the least y-value, we must first find where the minimum of the function is. This is achieved by finding the derivative and then testing values. The derivative of this function is

We then need to factor that and set it equal to 0, which gives us .

We then set each expression equal to 0 to give us our critical points.

This give us critical points at . We then set up a number line and test values on each side of the values. To the left of , you can choose  and plug that into the derivative. We get a positive value. In between the two critical points, you can choose , which gives us a negative value. To the right of , you can choose , which gives a positive value. To find the minimum, you must find the point where the sign changes from negative to positive. That happens at . That is the x-value of the minimum.

To find the y-value, you must plug that x-value into the original function: 

.   

Example Question #10 : How To Find Local Minimum Graphing Functions Of Curves

Given a graph with an equation  find the local minimum(s) if any are present.

Possible Answers:

No local minima.

Correct answer:

No local minima.

Explanation:

In order to solve this equation, we must first underestand that by taking the derivative of an equation of a graph and setting it equal to zero, we can find the values of  where there are critical points.  Critical points are either local maxima or minima, in order to figure out which you simply plug in numbers before and after that value of  in order to see whether or not the slope increases/decreases as is approaches or leaves that  value.  

In order to take the derivative of equation, the power rule must be applied, .   Taking the derivative of the graph equation, we find that it is .

Setting the equation equal to zero and solving for , we find that the critical points for this graph are present at .  Now we must determine whether these critical points are local minima or maxima.  

 

In order to determine whether or not they are local minima/maxima, we must determine the slope behavior of the graph around these points.  If the slope is positive towards a critical point and negative away from that critical point, that critical point is a local maxima.  Vice versa for local minima.   is approximately 1.38.   is approximately -1.38.  Therefore we can plug in -2,0,2 into the slope equation in order to determine the behavior of the slope around those points.

 

Plugging  into the derivative of the graph equation, we find that slope is positive.

Plugging  into the derivative of the graph equation, we find that the slope is positive as well.

Plugging  into the derivative of the graph equation, we find that the slope is positive.

 

Because the slope is always increasing, this means that there are no local minima in this graph.

 

Example Question #11 : How To Find Local Minimum Graphing Functions Of Curves

Find the local minima for the following function on the interval :

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

To find the local minima of a function, we must find the point(s) at which the first derivative changes from negative to positive.

The first derivative of the function is:

and was found using the following rule:

Now, we must find the critical points, the points at which the first derivative is zero:

Now, we create intervals over which to see whether the first derivative is positive or negative:

On the first interval, the first derivative is negative, while on the second interval it is positive, indicating that the point of the change in sign  is the local minimum.

Example Question #12 : How To Find Local Minimum Graphing Functions Of Curves

What is the minimum of over the interval ?

Possible Answers:

Correct answer:

Explanation:

To find the minimum of a function, find the first derivative. In order to find the derivative of this fuction use the power rule which states, .

Given the function,  and applying the power rule we find the following derivative.

Check the -value at each endpoint and when the first derivative is zero, namely

The smallest value is .

Example Question #13 : How To Find Local Minimum Graphing Functions Of Curves

Find the -value where the local minimum occurs on

.

Possible Answers:

Correct answer:

Explanation:

To find the minimum of a function, find the first derivative. In order to find the derivative of this fuction use the quotient rule which states, 

.

Given the function,  and applying the quotient rule we find the following derivative.

 when  and  when , which indicates that has a local minimum at .

Example Question #14 : How To Find Local Minimum Graphing Functions Of Curves

Find the relative minimum of .

Possible Answers:

x

Correct answer:

Explanation:

To find the relative minimum, you must first find the derivative of the function so that you can find the critical points. When taking the derivative, multiply the exponent by the leading coefficient and then subtract 1 from the exponent: . Then, set that equal to 0 to get your critical points: . Then, test on either side of that point to see what the behavior of the function is. To the left of 2 (plugging in 0, for example), the value is negative. To the right, the value is positive. Since the slope of the function is going from negative to positive at x=2, there is a relative minimum at x=2.

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