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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 543

Please help me in this question

Question

Determine the interval in which f is increasing or decreasing for the domain mentioned

F(x)=sin(x) x belongs (-pi,pi)

2nd question

And f(x)=cos(x) x belongs to (-pi÷2,pi÷2)

Math !!!!.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 8,006

hi Zeeshan 01

If you draw the graph of a function, increasing means those sections of the graph where the curve is going upwards, and decreasing means the curve is going downwards.

You may be able to answer these questions just by sketching the graphs. Or you could differentiate to get the gradient function and then consider: when is the gradient function positive and when is it negative?

There is a really useful grapher at this MathisFun site: http://www.mathsisfun.com/data/function-grapher.php

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 543

Plese show me method of differntiate

Math !!!!.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 8,006

Differentiation is a huge topic. Whole books have been written about it! You can make a start here:

http://www.mathsisfun.com/calculus/deri … ction.html

But you can avoid this by making the graphs.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 543

i perform all calculatiions in differntiation but this topic i dont understand

Math !!!!.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 8,006

Here is a graph of a function. The green curves show where it is increasing. The red curves show where it is decreasing.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**CIV****Member**- Registered: 2014-11-09
- Posts: 66

The derivative of sin x is cos x. Derivatives are equations used to calculate the SLOPE of a tangent line at some point on a curve. The curve is a function such as X^2 or in your example sin x. The derivative of X^2 is 2x. 2x is a linear equation and therefore is refereed to as a SLOPE equation. You can use that slope equation to calculate the ACTUAL slope of a tangent line on the curve of X^2 by choosing a point within the domain of the curve. Lets choose x = 2. The slope would be 2(2) = 4. The slope is represented with the letter m, so m = 4. Now that you have the slope, you need a y value to go with your x value of 2. Use that x value with the original equation to get yourself a y value: (2)^2 = 4. So now you have yourself a the slope of a tangent line at the point (2,4) on the curve X^2. Now what you need is the equation of this tangent line at that point. You remember the point-slope formula? Plug in all the information you have and solve for the equation for that tangent line: y - y2 = m(x-x2); y - 4 = 4(x - 2); y = 4(x - 2) + 4 = 4x - 8 + 4 = 4x - 4. So you have a point (2, 4); a slope equation of f'(x) = 2x; a slope of m = 4; and an equation of a tangent line, y = 4x - 4, at that point.

That's a basic derivative explanation. Forgive me if it appears that I'm assuming you know nothing.

It works the same for the function sin x. The slope equation sin x is f'(x) = cos x. One thing to keep in mind is that the graph of sin x goes to infinity to both direction. It repeats, goes round and round the unit circle. This is why your given an interval of (-pi, pi).

Now, it says determine when f is increasing and decreasing. A straight line is increasing when it's slope is positive and is decreasing when it's slope of negative. So if m = -2, that line is decreasing and if m = 4, that line is increasing. Positive increasing, negative decreasing.

So you have a function sin x and a slope equation for that function, which is cos x. Do you know you know how to find minima and maxima? The slope of minima and maxima is zero, m = 0. So now you know how to find minima and maxima. You set the first derivative, aka the slope equation for the function sin x, to zero. So now that cos x = 0, you have to ask yourself when is cos x zero? Cos x is zero at pi/2 and 3pi/2.

You might be asking why you have to find minima and maxima? By finding minima and maxima, you'll know where to check for increasing and decreasing slopes.

So now that you know that cos x = 0 at pi/2 and 3pi/2, are these values within your interval? pi/2 is, but 3pi/2 is not. If pi/2 is within the interval (-pi, pi), then so is -pi/2.

So you have these two points now. At these points cos x = 0 and being that cos x is the slope equation for sin x... the points on sin x with a slope of zero are the minima and maxima, the high and low points.

Now that you know where the high and low points are on sin x, those being -pi/2 and pi/2, you now know where to look for increasing and decreasing. You look between the high and low points, the minima and maxima. So you need a point between 0 and pi/2, and pi/2 and pi. Let's chose pi/4 and 3pi/4. If pi/4 and 3pi/4 are in the interval (-pi, pi) then so is -pi/4 and -3pi/4.

Now that we have points between -pi, -pi/2, pi/2, and pi; we can check for increasing and decreasing. Just use your calculator for this. f'(pi/4) = cos (pi/4) = 0.71, is that increasing or decreasing? Its increasing. Now we check f'(3pi/4) = cos (3pi/4) = -0.71, is that increasing or decreasing? It's decreasing. Do that rest and you we see whats happening.

Now remember! SIN X is the FUNCTION or CURVE. COS X is the DERIVATIVE or SLOPE EQUATION for a tangent line touching the function sin x. 0.71 and -0.71 are SLOPES of the tangent lines, the slope being the variable m, at the points pi/4 and 3pi/4.

If you want, you can solve for the equations of the tangent lines that same way I did in the very first explanation. Solve sin x using your x values to get y values. Using those y values, the x values, and the slopes, you can use the point-slope formula to solve for the tangent line equations.

I hope this helps. I can't believe I spent this much time writing all this out....

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 543

I don't understand it well !!!!

Please explain this question

F(x)=sin(x) x belongs (-pi,pi)

Math !!!!.

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 543

How we get this?????

pi/4 and 3pi/4.

Math !!!!.

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