Finding Gradients of Curves
Introduction
I am going to investigate the gradients of different curves and try to
work out a pattern that I could use to find the gradient of any curve.
I will draw graphs of a selection of curves, some by hand, some using
Autograph and some using Excel. I will use three methods to
investigate the graphs. Firstly, I will draw tangents to the curves at
4 or 5 points and measure the gradients. Secondly, I will draw chords
between x = 1 and 4 or 5 points and measure the gradients. Thirdly, I
will use algebra to work out a formula for the gradient and see how
this matches the first two methods.
At first I split up the coursework into 3 main families (for each
family there are additional equations to investigate):
Part One: Curves involving x2
1. y = x2
2. y = 2x2
3. y = 3x2
4. y = 4x2
5. y = x2 + 1
6. y = 7x2 + 6
Part Two: Curves involving x2 + x
1. y = x2 + x
2. y = x2 + 2x
3. y = 7x2 + 4x + 5
Part Three: Curves involving x3 + x
1. y = x3
2. y = 2x3
3. y = 4x3 + 2x - 5
Finally, I will summarise my results in a series of tables and work
out an overall formula that I could use to predict the gradient of any
curve.
PART ONE: CURVES CONTAINING X2
(1) y = x2
I am investigating the changes in gradient for the curve y = x2. To
plot the curve, I will use the table of values given below.
x
0
1
2
3
4
5
6
y
0
1
4
9
16
25
on the y. If my prediction is right I should be able to draw a
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I have plotted graphs from both sets of calculated gradients however I will concentrate on the graph plotted from the results show above as
Read the initial buret readings for both burets to the nearest 0.01 ml. Use a buret reading card to make the meniscus more prominent. Record readings on the report sheet. Have your instructor check and initial your report sheet for your first buret reading (sample #1, only). 6. Rinse a clean 125 ml Erlenmeyer flask with deionized water. Deliver approximately 20 ml of unknown acid into the Erlenmeyer flask. The tip of the buret should be approximately 1/2 inch below the top of the flask to avoid loss due to splashing. 7. Add 2 or 3 drop of phenolphthalein indicator. (Above your lab bench). 8. Titrate the unknown acid by adding standard NaOH (from the buret). Swirl the flask to mix the solutions during the addition of base. As the base is added you will observe a pink color localized at the spot the NaOH enters the solution (this is due to a localized high base concentration). Occasionally, rinse down the walls of your flask with deionized water (This rinses down any acid that has splashed onto the walls of your flask). Near the end-point, the pink color "flashes" throughout the solution and remains for a slightly longer time (1-2 seconds). When this occurs, add the NaOH drop by drop and eventually half-drops until the pink color remains (for at least 30 seconds). This is the end-point! NOTE: If you over-shoot the end-point (too much NaOH is added), add 1-2 more ml of the Unknown acid and then add NaOH again until a proper end-point is reached. Be sure
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