When seeing if a function is increasing or decreasing, while finding the domain, you are usually working the First Derivative Test. To reach this you follow these steps: 1. Take derivative, and set equal to 0. 2. Find anywhere it is not differentiable. 3. Set up intervals using steps 1 & 2. 4. Plug in a number on the interval into the derivative equation. 5. If positive, it is increasing, if negative it is decreasing. 6. Determine max or mins from step 5, increasing/decreasing > max decreasing/increasing > min Example: Find the open intervals on which f(x) = x^3 - 3/2x^2 is increasing or decreasing. 1. 3x^2 - 3x = 0 3x(x-1) = 0 x = 0, 1 2. ok 3. (-infiniti, 0)u(0, 1)u(1, infiniti) 4. f(-1) = neg. f(1/2) = neg. f(2) = pos. 5. f(-1) > decreasing f(1/2) > decreasing f(2) > increasing 6. x = 0 > nothing; x = 1 > min. DECREASING: (-infiniti, 0)u(0, 1) INCREASING: (1, infiniti)
The domain is all "x" values for which the graph is valid. For a lot of graphs, it's negative infinity to positive infinity. So for example any line, such as:
y = 2x + 7
can take any possible x value, positive or negative. The domain is then all x numbers, or you can say the domain is negative infinity to positive infinity. But something like:
y = sqrt(x)
can not take all numbers, because the square root of a negative number is not defined (as real). Therefore, the domain of this function is x >=0, or you can say that the domain is 0 (inclusive) to infinity.
You can also have cases where you just exclude certain numbers- so for example:
y = 1/x
is defined at all numbers EXCEPT 0. So then the domain is all numbers except 0, or you could say it's negative infinity to 0 (not inclusive) and 0 (not inclusive) to infinity.
So to find the domain, just ask yourself what values x can take. Essentially, just remember that you can't have a negative inside a square root and you cannot divide by 0. Those are the two main rules. Exclude those values and keep the rest.One of the best ways to do this actually is to graph the function and visually inspect it. However, if you do this, just make sure that there isn't an issue outside of your viewing area.
When seeing if a function is increasing or decreasing, while finding the domain, you are usually working the First Derivative Test. To reach this you follow these steps:
ReplyDelete1. Take derivative, and set equal to 0.
2. Find anywhere it is not differentiable.
3. Set up intervals using steps 1 & 2.
4. Plug in a number on the interval into the
derivative equation.
5. If positive, it is increasing, if negative
it is decreasing.
6. Determine max or mins from step 5,
increasing/decreasing > max
decreasing/increasing > min
Example:
Find the open intervals on which
f(x) = x^3 - 3/2x^2 is increasing or decreasing.
1. 3x^2 - 3x = 0
3x(x-1) = 0
x = 0, 1
2. ok
3. (-infiniti, 0)u(0, 1)u(1, infiniti)
4. f(-1) = neg.
f(1/2) = neg.
f(2) = pos.
5. f(-1) > decreasing
f(1/2) > decreasing
f(2) > increasing
6. x = 0 > nothing; x = 1 > min.
DECREASING: (-infiniti, 0)u(0, 1)
INCREASING: (1, infiniti)
The domain is all "x" values for which the graph is valid. For a lot of graphs, it's negative infinity to positive infinity. So for example any line, such as:
ReplyDeletey = 2x + 7
can take any possible x value, positive or negative. The domain is then all x numbers, or you can say the domain is negative infinity to positive infinity. But something like:
y = sqrt(x)
can not take all numbers, because the square root of a negative number is not defined (as real). Therefore, the domain of this function is x >=0, or you can say that the domain is 0 (inclusive) to infinity.
You can also have cases where you just exclude certain numbers- so for example:
y = 1/x
is defined at all numbers EXCEPT 0. So then the domain is all numbers except 0, or you could say it's negative infinity to 0 (not inclusive) and 0 (not inclusive) to infinity.
So to find the domain, just ask yourself what values x can take. Essentially, just remember that you can't have a negative inside a square root and you cannot divide by 0. Those are the two main rules. Exclude those values and keep the rest.One of the best ways to do this actually is to graph the function and visually inspect it. However, if you do this, just make sure that there isn't an issue outside of your viewing area.