In this lesson,
we will use the free online graphing calculator linked below
to graph the function, adjust the window,
then determine derivative function values.
We are given f
of x equals one third x cubed minus 4x plus two
and we're asked to find f prime of zero, f prime of two
and f prime of negative three.
Remember, the derivative function value is the slope
of the tangent line at a given x value, which we will verify
once we find these derivative function values.
To begin, we go to the graphing calculator
and enter the given function.
For this calculator, it's important to recognize
that we find derivative function values using
the graphing feature of the calculator,
not the calculation window, below.
Let's clear the existing functions.
And enter the given function, f
of x equals one third x cubed
minus 4x plus two.
The character exponent key from the keyboard is Shift 6.
Next let's click Plot Graphs.
Let's adjust the window,
so we have a better view of the graph.
We can do this several ways.
One way is to adjust the window manually by clicking
on the Window tab and adjusting the max
and min values for the x and y-axes.
Another way though would be to zoom out
or zoom in, using the small plus or minus buttons
in the lower right-hand corner of the graph.
We can also shift the graph left, right, up and down.
Let's go and try zooming out by clicking the minus.
Now we have a nice view of the graph.
Let's go and shift the graph down by clicking the up button.
And now we have a nice view of the function.
To calculate the derivative function values
at x equals zero, x equals two
and x equals negative three, we click the Calculation tab,
in the upper left.
Click the dropdown menu next to Mode and select Derivative.
Next we're asked to click on the curve.
From here we simply enter the x values
where we want the derivative.
We want f prime of zero, so we leave the value of x at zero.
Click Calculate and the derivative function value
at x equals zero is equal to negative four, meaning f prime
of zero is equal to negative four.
Notice how the information is also recorded below
in the calculation window.
Next you want to find f prime of two,
so we change the x value to two.
Click Calculate, f prime of two is equal to zero.
And finally, you want f prime
of negative three, we change x value to negative three.
Click Calculate.
f prime of three is equal to five.
Let's go ahead and record these derivative function values
and then we'll verify they are the slopes
of the tangent lines, at the given x values.
f prime of zero is equal to negative four.
f prime of two is equal to zero
and f prime of negative three is equal to five.
And again, this verify these do represent the slopes
of the tangent lines at the given x values.
f prime of zero indicates the slope of the tangent line
at x equals zero is equal to negative four.
Going to the graph, here's the point
on the function where x equals zero.
If we sketch the tangent line,
the slope of the black tangent line is negative four.
f prime of two equals zero indicates the slope
of the tangent line is zero when x is equal to two.
The point on the graph where x equals two is
this point here which represents a relative minimum.
If we sketch a tangent line
at that point, we do have a horizontal line where the slope
is equal to zero.
And the finally, f prime
of negative three equals five indicates that the point
on the graph where x equals negative three,
which is this point here, the slope
of the tangent line is five.
And if we sketch the tangent line,
the slope is positive five.
I hope you found this helpful.