Here is the link to the graph of f' in question:
bit.ly/2oml70V
I have a hard time trying to answer these kinds of questions when we're only given a graph of f'.
a) Approximate the slope of f at x = 4. Explain.
I believe the slope of f in this case would be a decreasing slope?
b) Is it possible that f(2) = -1? Explain.
This one I'm not quite sure. I think it is possible but not entirely sure on how to explain that one.
c) Is f(5) - f(4) > 0? Explain.
d) Approximate the value of x where f is maximum. Explain
e) Approximate any open intervals in which the graph of f is concave upward and any open intervals in which it is concave downward. Approximate the x-coordinates of any points of inflection.
Any help is greatly appreciated!
Any ideas? Really need help on this one
(a) the graph is f', which is the slope of f. So, the slope at x=4 is on the graph, and appears to be -1
(b) sure it's possible. f(x) = ?f'(x) dx = "f(x)" + C
where "f(x)" is the antiderivative of f'(x)
So, f(2) can be anything you want, depending on the value of C
(c) since f'(x) < 0 on [4,5], it is decreasing. So, f(5) < f(4)
(d) f is maximum where f'(x) = 0 and f"(x) < 0 (concave down).
f"(x) < 0 where f' is decreasing. So, that means that f(3.5) is a maximum.
(e) f is concave up where f" > 0
That is where f' is increasing, or (-?,1)U(5,?)
f is concave down where f" < 0, or (1,5)
To check these answers, consider that a reasonable for f'(x) is
f'(x) ? (x+0.8)(x-3.5)(x-6.25)
so take a look at the graph at
http://www.wolframalpha.com/input/?i=integral+(x%2B0.8)(x-3.5)(x-6.25)
and you will see that it fits the above answers. In fact, with C -45.28, f(2) = -1
a) To approximate the slope of f at x = 4, we can look at the graph of f'. The slope of f at x = 4 corresponds to the value of f'(4) on the graph. Look at the point on the graph where x = 4, and find the value on the y-axis. This value represents the slope of f at x = 4. If the graph of f' is decreasing at x = 4, then the slope of f at x = 4 would be negative.
b) To determine if it is possible that f(2) = -1, we need to look at the graph of f'. The value of f(2) on the y-axis represents the height of the graph of f at x = 2. If the value on the y-axis is -1, then it is indeed possible for f(2) = -1. However, without the actual graph, we cannot say for certain.
c) To determine whether f(5) - f(4) > 0, we need to look at the graph of f'. The difference between f(5) and f(4) can be thought of as the signed area under the graph of f' between x = 4 and x = 5. If the area under the graph is positive, then f(5) - f(4) > 0. If it is negative, then f(5) - f(4) < 0. If it is zero, then f(5) - f(4) = 0.
d) To approximate the value of x where f is maximum, we need to look for the highest point on the graph of f'. This corresponds to the x-value where f' changes from positive to negative (or from increasing to decreasing). It would be helpful to observe the graph and identify this point visually.
e) To find intervals where the graph of f is concave upward or concave downward, we need to look for regions on the graph of f' where the slope of f' is increasing or decreasing, respectively. Concave upward regions correspond to increasing slopes of f', and concave downward regions correspond to decreasing slopes of f'. Points of inflection occur where the concavity changes (i.e., where the slope of f' changes from increasing to decreasing or vice versa). Again, having access to the actual graph of f' would greatly aid in approximating these intervals and points of inflection.
To answer these questions based on the given graph of f', we can use the properties of derivative and graph analysis. Let's go through each question step by step.
a) To approximate the slope of f at x = 4, we can look at the graph of f' and observe the behavior around x = 4. The slope of f at a specific point can be found by analyzing the value of f' at that point. In this case, locate x = 4 on the graph of f' and see the corresponding value of f'.
If the graph of f' is decreasing as x approaches 4, then the value of f' at 4 would be negative. This indicates that the slope of f at x = 4 is negative.
b) To determine if f(2) = -1 is possible, we need to consider the relationship between f and its derivative, f'. We know that the derivative represents the rate of change of f.
If f(2) = -1, it means that at x = 2, the function f has a tangent line with a slope of -1. So to verify if this is possible, locate x = 2 on the graph of f' and observe the corresponding value of f'.
If the value of f' at x = 2 matches -1, then it is possible for f(2) = -1.
c) To determine whether f(5) - f(4) > 0, we can analyze the graph of f'. The difference between f(5) and f(4) represents the change in the function f between x = 4 and x = 5.
By examining the graph of f' between x = 4 and x = 5, observe whether the graph is located above the x-axis or below. If the graph of f' is located above the x-axis in this interval, then f(5) - f(4) would be positive. Otherwise, if the graph is located below the x-axis, then the difference would be negative.
d) To approximate the value of x where f is maximum, we need to look for an extremum point on the graph of f'. The extremum points (maximum or minimum) on f' correspond to the x-values where f has a maximum or minimum.
Examine the graph of f' and find any peaks or valleys. Locate the corresponding x-values and those would be the approximate values of x where f has a maximum or minimum.
e) To determine the intervals where the graph of f is concave upward or downward, we need to analyze the concavity of f' using its graph.
An upward concave graph of f' indicates that the graph of f is concave upward. Locate any such intervals on the graph of f' and note the corresponding x-values.
Similarly, a downward concave graph of f' indicates that the graph of f is concave downward. Locate any such intervals on the graph of f' and note the corresponding x-values.
To approximate the x-coordinates of any points of inflection, examine the points on the graph of f' where the concavity changes from upward to downward or vice versa. These points of change in concavity correspond to points of inflection on the graph of f.
By following these steps and analyzing the given graph of f', you can determine the answers to the different questions.