Name: _____________________
Geol 351- Geomathematics
Derivatives warm-up. Look over for next time (March 14)
In-class problem
1. In the porosity density relationship = 0e(-z/), assume 0 is 0.4 and  is 1.5. What is the
slope (or porosity gradient) between depths of 1 and 2 km in this area? What is the slope
between 1.4 and 1.6 kilometers? Calculate these slopes out explicitly using the relationship
2  1

slope 
t

z2  z1
. Sketch your results on the graph of this function located on the back of this
page.
Use the space below for calculations as needed.
Hand in end of class March 14
Draw a line through the points on the curve at depths of 1.4 and 1.6. Extend the line so that it crosses
the z=1 and 2 grid lines (so z=1 to facilitate a quick graphical computation). What is the difference
in  and compare the slope (/z) of this tangent line to your computations on the preceding page.
Pick a couple additional points at, for example z=0.5 and 3, use a ruler and connect the points to get
the tangent line. How do the variations in the slopes of these tangent lines compare to the function
(z)?
Porosity depth relationship
0.4
-z/1.5
=0.4e
0.3
PHI
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
1
2
3
4
5
Z
Tear off and had in end of class March 14
Geol 351- Geomathematics
Some additional in-class work
Calculate the slope of the cosine function at =0, /4 and /2 radians. Plot your results on the graph
(next page). Calculate the slope using two points located on either side of these values. You could
use a  of 0.01 or 0.02 radians. If you wish, use Excel to make these calculations. As you make
these calculations, look at the graph and ask yourself if they make sense.
Use next page for calculations if needed
Calculate and show your results below.
cos  2  cos 1 cos 0.01  cos 0.01 0.999  0.999
At 0 radians the slope equals


0
 2  1
0.01  (0.01)
0.02
At /4 (0.785) radians, the slope equals
cos  2  cos 1 cos(0.79)  cos(0.78) 0.704  0.711 0.007



 0.7
2  1
0.79  0.78
0.01
0.01
At /2 (1.571) radians, the slope equals
cos  2  cos 1 cos(1.581)  cos(1.561) 0.01  0.01 0.02



 1
2  1
1.581  1.561
0.02
0.02
At 3/4 (2.356) radians, the slope equals
cos  2  cos 1 cos(2.366)  cos(2.346)



 __________
2  1
2.366  2.346
0.02
0.02
Cosine function
1.0
0.0
-0.5
 (radians)
6.
0
5.
5
5.
0
4.
5
4.
0
3.
5
3.
0
2.
5
2.
0
1.
5
1.
0
0.
5
-1.0
0.
0
cos ()
0.5
Sketch the derivative of the cos()
1.0
d cos ()/d
0.5
0.0
-0.5
0
5
5.
6.
0
5.
0
4.
5
5
3.
4.
0
3.
2.
5
2.
0
5
1.
0
1.
5
0.
0.
0
-1.0
 (radians)
Plot the slopes computed on the previous page on the graph above. Based on those computations
sketch in the remainder of the curve and identify this trigonometric function.
The curve formed by the slopes (derivative) computed for the cosine looks like the
_______________ .
For discussion next time (March 14)
Geol 351- Geomathematics
Additional in-class work
Calculate the slope of the function y=x2 (see graph below). Plot your results on the graph (next
page). Calculate the slope at three points and connect the dots. If you wish, use Excel to make these
calculations. Confirm that the resultant line in fact equals the result you would obtain by taking the
derivative: dy/dx.
Use next page for calculations if needed
Calculate and show your results below.
For example, the slope at x=1, equals 0.4/0.2 ____________ (i.e. at x= 0.9 and 1.1, y=0.81 and 1.21
respectively: y  2.
x
At x=3, the slope equals 1.2/0.2 ____________
At x=5, for example, the slope equals ____________
Calculate the slope at – at least – one additional point and connect the dots.
On the next page write down the equation for the line fit to those three points
2
y=x
40
35
30
25
y 20
15
10
5
0
0
1
2
3
x
4
5
6
For discussion next time (March 14)
2
Add the slope of y=x
40
35
30
25
y 20
15
10
5
0
0
1
2
3
x
4
5
6
Plot the slopes you just computed.
The curve formed by the slopes computed for y=x2 looks like a _______________ .
What is the equation of this line?
What is the derivative of y=x2? _________________
Does the function defining the derivative equal the slope? _____________.
For discussion next time (March 14)