By what transformation can the set representing the inverse of a function be found?
1 answer:
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Answer:
x-intercept = 0.956
Step-by-step explanation:
You have the function f(x) given by:
(1)
Furthermore you have that at the point (a,f(a)) the tangent line to that point has a slope of -1.
You first derivative the function f(x):
(2)
To solve this derivative you use the following derivative formula:
For the derivative in (2) you have that b=2 and u=2x. You use the last expression in (2) and you obtain:
You equal the last result to the value of the slope of the tangent line, because the derivative of a function is also its slope.
Next, from the last equation you can calculate the value of "a", by doing x=a. Furhtermore, by applying properties of logarithms you obtain:
With this value you calculate f(a):
Next, you use the general equation of line:
for xo = a = 0.235 and yo = f(a) = 0.721:
The last is the equation of the tangent line at the point (a,f(a)).
Finally, to find the x-intercept you equal the function y to zero and calculate x:
hence, the x-intercept of the tangent line is 0.956
Part A) x-intercepts simply show that when the value of the function is zero. Vertex coordinates show that when the function obtains its maximum value. When x=50, function obtains its maximum value and it's 75. The function is increasing in the interval (0, 50) and decreasing in the interval (50, 100). In regard to the height and distance of the tunnel, these numbers show that decreasing and increasing intervals are symmetric. Each number from the intervals has its own pair in the corresponding interval and they are located in the same distance from the midpoint (50,75)
Part B) In order to calculate the average rate of change, we can first write the function. Using the information about the x-intercept and the vertex coordinates, we find that our function is
.
Plugging 15 and 35 in x, we can find the values of the function, i.e.
and 
.
Then, the average change is
Part B) In order to calculate the average rate of change, we can first write the function. Using the information about the x-intercept and the vertex coordinates, we find that our function is
Plugging 15 and 35 in x, we can find the values of the function, i.e.
Then, the average change is