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
Do it for me please...
I have a hunch you actually mean:
g(x)=x/ (x-7)
because that is much more exciting than what you typed.
Now clearly if x = 7 we have a serious problem with this function. It has a zero for its denominator and is therefore undefined at x = 7.
Now what if x = 8 ?
g(8) = 8/1 = 8
And what if x = 6?
g(6) = 6/-1 = -6
Sketch a graph now and watch what is happening
try x = 20
g(20) = 20/13
try x = -20
g(-20) = -20/-27
try x = 100
g(100) = 100/93 humm closer and closer to 1
try x = -100
g(-100) = -100/-107 humm, closer and closer to -1
Now do the x = 7.5 and 6.5 like they said
Now slopes.
You did not say if you know calculus. If not brute force or just use your graph. If you do know then the slope is the derivative at that point so
g'(x) = [(x-7) -x ] / (x-7)^2
or
slope = -7/(x-7)^2
try x = -100
g(-100) = -100/-107 humm, closer and closer to +1
by the way, notice that the slope is negative everywhere.
To summarize the increasing and decreasing intervals of the function g(x) = x/(x-7), we need to determine the values of x for which the function is increasing or decreasing.
a) To find the increasing and decreasing intervals:
1. Identify the critical points. These occur when the derivative of the function is equal to zero or undefined. In this case, the derivative of g(x) is:
g'(x) = (x-7 - x)/((x-7)^2) = -7/((x-7)^2)
2. Find the values of x at which the derivative is zero or undefined. Since the denominator cannot be zero, the derivative is undefined when x = 7.
3. Now, determine the sign of the derivative in the intervals (x < 7) and (x > 7) using a test value in each interval. For instance, pick x = 6 (in (x < 7)), and substitute it into the derivative:
g'(6) = -7/((6-7)^2) = -7
Since the derivative is negative, the function is decreasing in the interval (x < 7).
4. Similarly, pick x = 8 (in (x > 7)) and substitute it into the derivative:
g'(8) = -7/((8-7)^2) = -7
The derivative is still negative, indicating that the function is also decreasing in the interval (x > 7).
Therefore, the function g(x) = x/(x-7) is decreasing for all x values.
b) Comparing the slopes of the tangents at the given points:
i) To find the slope of the tangent at x = 7.5, calculate the derivative g'(x) and substitute x = 7.5:
g'(7.5) = -7/((7.5-7)^2) = -7/0.25 = -28
Thus, the slope of the tangent at x = 7.5 is -28.
Next, calculate the slope of the tangent at x = 20:
g'(20) = -7/((20-7)^2) = -7/169
Therefore, the slope of the tangent at x = 20 is -7/169.
ii) For x = 6.5, calculate the derivative g'(x) and substitute x = 6.5:
g'(6.5) = -7/((6.5-7)^2) = -7/0.25 = -28
Hence, the slope of the tangent at x = 6.5 is -28.
Similarly, for x = -20:
g'(-20) = -7/((-20-7)^2) = -7/729
Therefore, the slope of the tangent at x = -20 is -7/729.
In summary:
- The slopes of the tangents at x = 7.5 and x = 6.5 are both -28.
- The slopes of the tangents at x = 20 and x = -20 are -7/169 and -7/729, respectively.