Consider the function g(x)=x/x-7.
a) Summarize the increasing and decreasing intervals.
b) Compare the slopes of the tangents at the points where i) x=7.5 and x=20
ii) x=6.5 and x=-20
done below:
http://www.jiskha.com/display.cgi?id=1386792229
ya i seen that on have the same question that i got, but when i try to sketch the function of x/x-7 on the wolfram Alpha i get the wrong answer, so i couldn't do a and b without graphing this.
*help me please*
probably did not get the correct graph because you skipped the brackets
If you entered the equation as
y = x/x-7 , you got a straight horizontal line at y = -6
because
y = x/x - 7 = 1-7 = -6
You MUST include brackets, such as
y = x/(x-7)
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%2F%28x-7%29+%2C+y+%3D+x%2Fx+-+7
The graph in red would be yours,
the one in blue would be mine.
Thank you so much Reiny. thanks a lot.
I couldn't do g). someone help i need it in the morning. help me please. do it for me please. Show me the steps too please...please...
i meant b) please help me....
To summarize the increasing and decreasing intervals of a function, we need to analyze the sign of the derivative of the function. The derivative describes the rate at which the function is changing at each point.
First, let's find the derivative of the function g(x) = x / (x - 7). We can use the quotient rule to differentiate it. The quotient rule states that if we have a function in the form f(x) / g(x), its derivative can be found as (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2.
Applying the quotient rule to g(x), we have:
g'(x) = [(x - 7)' * (x) - (x)' * (x - 7)] / (x - 7)^2
Simplifying further, we get:
g'(x) = [(1) * (x) - (1) * (x - 7)] / (x - 7)^2
= 7 / (x - 7)^2
Now, let's consider the intervals of increasing and decreasing.
a) Increasing and Decreasing Intervals:
To determine when the function is increasing or decreasing, we need to analyze the sign of the derivative.
Since g'(x) = 7 / (x - 7)^2, the derivative is always positive (7 is positive, and (x - 7)^2 is always positive) except when x = 7.
Therefore, g(x) is increasing for all values of x except when x = 7, and it is decreasing only when x = 7.
b) Comparing the Slopes of Tangents at different points:
To compare the slopes of the tangents at different points, we need to find the derivative at those points.
i) x = 7.5 and x = 20:
Plugging these values into the derivative equation, we get:
g'(7.5) = 7 / (7.5 - 7)^2
= 7 / (0.5)^2
= 7 / 0.25
= 28
g'(20) = 7 / (20 - 7)^2
= 7 / (13)^2
= 7 / 169
The slope of the tangent at x = 7.5 is 28, and the slope of the tangent at x = 20 is 7 / 169.
ii) x = 6.5 and x = -20:
Plugging these values into the derivative equation, we get:
g'(6.5) = 7 / (6.5 - 7)^2
= 7 / (-0.5)^2
= 7 / 0.25
= 28
g'(-20) = 7 / (-20 - 7)^2
= 7 / (-27)^2
= 7 / 729
The slope of the tangent at x = 6.5 is 28, and the slope of the tangent at x = -20 is 7 / 729.
In summary:
i) The slope of the tangent at x = 7.5 is 28, and the slope of the tangent at x = 20 is 7 / 169.
ii) The slope of the tangent at x = 6.5 is 28, and the slope of the tangent at x = -20 is 7 / 729.