MHB Determine the Average Rate of Change

AI Thread Summary
The average rate of change (AROC) of the function y = 2cos(x - π/3) + 1 over the interval π/3 ≤ x ≤ π/2 was initially calculated incorrectly, yielding an approximate value of 1.4. Upon further analysis, the correct AROC was determined to be approximately -0.5117 after identifying errors in the calculations. The discrepancy was clarified by graphing the function and analyzing the secant line. This highlights the importance of careful computation and verification in determining the AROC. The final correct value reflects the function's behavior over the specified interval.
eleventhxhour
Messages
73
Reaction score
0
Determine the average rate if change of the function y = 2cos(x - $\pi$/3) + 1 for the interval $\pi$/3 $\le$ x $\le$ $\pi$/2

I tried finding the exact values of the two (0 and 0.5) and subbing them into the AROC equation but I keep getting the wrong answer (1.4)
 
Mathematics news on Phys.org
We find:

$$\frac{\Delta y}{\Delta x}=\frac{y\left(\dfrac{\pi}{2}\right)-y\left(\dfrac{\pi}{3}\right)}{\dfrac{\pi}{2}-\dfrac{\pi}{3}}=\frac{\left(2\cos\left(\dfrac{\pi}{6}\right)+1\right)-\left(2\cos\left(0\right)+1\right)}{\dfrac{\pi}{6}}=\frac{6\left(\sqrt{3}-1\right)}{\pi}\approx1.39811405542801$$
 
MarkFL said:
We find:

$$\frac{\Delta y}{\Delta x}=\frac{y\left(\dfrac{\pi}{2}\right)-y\left(\dfrac{\pi}{3}\right)}{\dfrac{\pi}{2}-\dfrac{\pi}{3}}=\frac{\left(2\cos\left(\dfrac{\pi}{6}\right)+1\right)-\left(2\cos\left(0\right)+1\right)}{\dfrac{\pi}{6}}=\frac{6\left(\sqrt{3}-1\right)}{\pi}\approx1.39811405542801$$

Okay, that's what I got. The textbook has it as -0.5157 so I guess it's just wrong?
 
We both made an error...it should be:

$$\frac{\Delta y}{\Delta x}=\frac{6\left(\sqrt{3}-2\right)}{\pi}\approx-0.511745261674736$$

I discovered my error when graphing the function and the resulting secant line:

View attachment 3547
 

Attachments

  • aroc.png
    aroc.png
    13.9 KB · Views: 96
Thread 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...

Similar threads

Replies
4
Views
1K
Replies
1
Views
2K
Replies
5
Views
3K
Replies
8
Views
3K
Replies
1
Views
2K
Replies
7
Views
2K
Replies
4
Views
1K
Back
Top