Compare the functions shown below: Which function has the greatest maximum y-value?
2 answers:
Answer: It's a tie between f(x) and h(x). Both have the same max of y = 3
The highest point shown on the graph of f(x) is at (x,y) = (pi,3). The y value here is y = 3.
For h(x), the max occurs when cosine is at its largest: when cos(x) = 1.
So,
h(x) = 2*cos(x)+1
turns into
h(x) = 2*1+1
h(x) = 2+1
h(x) = 3
showing that h(x) maxes out at y = 3 as well
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Note: g(x) has all of its y values smaller than 0, so there's no way it can have a max y value larger than y = 3. See the attached image to see what this graph would look like if you plotted the 7 points. A parabola seems to form. Note how point D = (-3, -2) is the highest point for g(x). So the max for g(x) is y = -2
The highest point shown on the graph of f(x) is at (x,y) = (pi,3). The y value here is y = 3.
For h(x), the max occurs when cosine is at its largest: when cos(x) = 1.
So,
h(x) = 2*cos(x)+1
turns into
h(x) = 2*1+1
h(x) = 2+1
h(x) = 3
showing that h(x) maxes out at y = 3 as well
--------------------------------
Note: g(x) has all of its y values smaller than 0, so there's no way it can have a max y value larger than y = 3. See the attached image to see what this graph would look like if you plotted the 7 points. A parabola seems to form. Note how point D = (-3, -2) is the highest point for g(x). So the max for g(x) is y = -2
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Using cross multiplication you can determine that the ANSWER is 10.
We can calculate this using arc length formula.
![L= \int\limits^a_b { \sqrt[]{1+ (\frac{dy}{dx} } )^2} \, dx](https://tex.z-dn.net/?f=L%3D%20%5Cint%5Climits%5Ea_b%20%7B%20%5Csqrt%5B%5D%7B1%2B%20%28%5Cfrac%7Bdy%7D%7Bdx%7D%20%7D%20%29%5E2%7D%20%5C%2C%20dx%20)
The first step is to find the derivative of the given function.
Now we plug this back into original integral.
![L= \int\limits^a_b { \sqrt[]{1+ 9x} \, dx](https://tex.z-dn.net/?f=L%3D%20%5Cint%5Climits%5Ea_b%20%7B%20%5Csqrt%5B%5D%7B1%2B%209x%7D%20%5C%2C%20dx%20)
We solve this integral using substituition u=9x+1. This way we end up solving elementary integral.
A final solution is:
Finaly we get that arc lenght is:
The first step is to find the derivative of the given function.
Now we plug this back into original integral.
We solve this integral using substituition u=9x+1. This way we end up solving elementary integral.
A final solution is:
Finaly we get that arc lenght is: