, and estimate what, f(1.01) would be without using a calculator.Mr. O'B gave us a trick, he said that since f is locally linear, to approximate it by using a tangent line equation. The tangent line is the derivative. To do this we tried to find the tangent line at x=1. It is close to 1.01, thus using the local linearity. We needed both a point and a slope.
We had half of what we needed, x=1, and we needed to find a y. We had a function to find the y-value, -2. Then it was time for slope. To find the slope we took the derivative. We didn't have nDeriv, but just used the new trick.
We just needed to make a linearization. If you want to learn more about linearization, click here.
New trick:
Slope:
- point slope of
is similar to , if x is near 1.
This graph shows both lines and how they are tangent to one another. As you zoom in on the point which we found, and put on the trace you can see that the lines look the same.If we wanted to make sure we were correct using algebra, we could say,
.We then talked about the power rule. Which says that
, where n can be any real number. The power rule works if all these things are true. We used the h-->0 derivative for this proof. The h-->0 is better for finding generic equations, whereas the other derivative set-up is better for finding specific answer, if you already have numbers.Proof: (for all real numbers) is explained on this link: click here.
When we did the proof in class we used the binomial theorem, which if you forgot it, is shown on this website:
click here.
We then went over the tricky problems on Assignment #1. The first question we spoke about was #4, which was the bird function problem. When you graphed this function it looked like a normal function, however, if you zoomed up very very close you could see that the function had a vertical tangent. You were taking a cubic and adding the cube root function to it. This brings us to the great point that if a function appears to have no cusp, be careful- you need to also see the equation. The function is not differential if there is a cusp.
#2 (on Assignment #1)-The graph shows that one of the functions has a cusp, it is not locally linear. The other is locally linear, therefor there is a derivative.
#5(on Assignment #1)-Was the absolute value, sin function graph. This shows when a graph is not differentiable at a corner.
From Assignment #1 we learned that graphs can lie to you. Equations are important because of this.
3 Places were a Derivative is not defined:
1. At a point of discontinuity
2. At a vertical tangent
3. At a corner/cusp
We then stated this theorem: If a function is differentiable(has a derivative) then it is continuous.
Problem #12 from the assignment asked if to find the derivative of
could you use the power rule. We tried to see if 
. To see if this was true we sketched y=2x on Geogebra and then find the derivative. To do this on the calculator you plug in 
. To do this on Geogebra you find the tangent line, by touching a point and getting a tangent line, the derivative is the sloope of the line. You plot the slope by graphing the x coordinate of that point, 
, then trace the point to see what happens. We found that was not the same was x2^(x-1). The power rule does not work for this function. The derivative of an exponential does not work for the power rule. My graph below shows how to find all of these steps on Geogebra.
We then looked at the Product Quotient Rule, which is explained on page 130. And the proof is on page 131. While doing this we learned a new term, FUFOZ, which stands for Funny Form of Zero.
If you want to see what we did you can check out example 8 on page 130. In class we stated the quotient rule but did not prove it.
We also checked out the proof to the sum/difference rule which is on page 128 in the text book. And the constant multiple proof is on page 129.
We are going to be finishing the notes on Monday.
HW: Assignment #2 on Blog, the HW part starts with #1, do that.
Update: This is a really cool video that helped me understand derivatives click here.
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