Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (If an answer does not exist, enter DNE.)
f(t) = 6 cos t, −3π/2 ≤ t ≤ 3π/2
I figured it out its -6 for absolute minimum and local minimum and 6 for absolute max and local max
excuse me? You don't know where the maxima and minima are on cos(x)?
Don't forget your trig now that you're doing calculus. If nothing else, just find f' and see where it's zero.
Is there some trick to this question beyond what you have posted?
Uhm I did take the derivative but the answers can only be between -3pi/2 and 3pi/2.
Sketch the graph like it said to do.
To sketch the graph of f(t) = 6 cos(t), we can start by considering the properties of the cosine function.
The cosine function oscillates between -1 and 1 as t increases.
Given that the domain of the function is -3π/2 ≤ t ≤ 3π/2, this corresponds to a range of -3π/2 ≤ f(t) ≤ 3π/2, which means that the graph of f(t) will oscillate between -6 and 6 along the y-axis.
To draw the graph, we can start by marking the points where f(t) = 6 cos(t) is at its maximum and minimum values.
The maximum value of cos(t) is 1, so the maximum value of 6 cos(t) occurs when cos(t) = 1. This happens at t = 0. Therefore, the point (0,6) will represent the maximum value on the graph.
Similarly, the minimum value of cos(t) is -1, so the minimum value of 6 cos(t) occurs when cos(t) = -1. This happens when t = π, and at t = -π. Therefore, the points (π,-6) and (-π,-6) will represent the minimum values on the graph.
Now, let's draw the graph based on these points.
The graph of f(t) will start at (0,6) and will reach its minimum values at t = π and t = -π, corresponding to the points (π,-6) and (-π,-6). The graph will then reach its maximum value again at (0,6).
The general shape of the graph will be a cosine wave that oscillates between these maximum and minimum values.
Therefore, the sketch of the graph will look like a wave that starts at (0,6), goes down to (-π,-6), comes back up to (0,6), and then continues to π,-6 in the same pattern.
Now, let's determine the absolute and local maximum and minimum values using the sketch.
- Absolute maximum value of f: The absolute maximum value of f occurs at the highest point on the graph. From the sketch, we can see that the highest point is (0,6). Therefore, the absolute maximum value of f is 6.
- Absolute minimum value of f: The absolute minimum value of f occurs at the lowest point on the graph. From the sketch, we can see that the lowest points are (-π,-6) and (π,-6). Therefore, the absolute minimum value of f is -6.
- Local maximum and minimum values of f: From the sketch, we can observe that there are no other points where the graph reaches a higher or lower value within any small interval around those points. Hence, there are no local maximum or minimum values.
To summarize:
- Absolute maximum value of f = 6.
- Absolute minimum value of f = -6.
- No local maximum or minimum values exist.