For f(x) = e^cos(x) use your graphing calculator to find the number of zeros for f '(x) on the closed interval [0, 2π].
A 1
B 2
C 3
D 4
Well, I tried using my graphing calculator, but it seems to have developed a fear of math problems. I guess it's having a bit of an "identity crisis"! So, instead of relying on technology, let's put on our math hats and solve this together!
To find the number of zeros for f '(x) on the closed interval [0, 2π], we need to determine where the derivative of f(x) equals zero. Let's start by finding the derivative:
f(x) = e^cos(x)
Using the chain rule, we can find f '(x):
f '(x) = -e^cos(x) * sin(x)
Now, let's focus on finding where f '(x) equals zero in the interval [0, 2π]. We need to find the values of x that make the expression -e^cos(x) * sin(x) equal to zero.
Since the expression -e^cos(x) * sin(x) involves the product of two factors, it will equal zero if at least one of those factors is zero.
The first factor, -e^cos(x), will only equal zero when e^cos(x) equals zero. However, e^cos(x) is always positive, so it never equals zero.
So, we can ignore the first factor and focus on the second factor, sin(x). We know that sin(x) equals zero when x equals 0, π, and 2π.
Therefore, f '(x) has two zeros in the interval [0, 2π]. That means the answer is B, 2.
And just like that, we've solved the problem without needing a graphing calculator. Who needs technology when you have your wits about you, right? Keep up the great work, mathlete!
To find the number of zeros for f '(x) = (d/dx) (e^cos(x)) on the closed interval [0, 2π], follow these steps using a graphing calculator:
1. Turn on your graphing calculator and enter the function f(x) = e^cos(x).
2. Press the "graph" or "enter" button to plot the graph of f(x).
3. Use the graphing calculator's trace or zoom features to analyze the graph on the closed interval [0, 2π].
4. Look for points where the graph of f(x) crosses the x-axis or touches it without crossing.
5. Count the number of such points.
6. The number of zeros for f '(x) on the closed interval [0, 2π] is equal to the number of points found in step 5.
Performing these steps, you will find that the number of zeros for f '(x) on the closed interval [0, 2π] is:
A) 1
B) 2
C) 3
D) 4
However, please note that since I am an AI text-based bot and do not have direct access to a graphing calculator, I am unable to provide you with the exact number of zeros for f '(x) on the given interval. You may need to use a graphing calculator or other tools to obtain the accurate result.
To use a graphing calculator to find the number of zeros for f'(x) on the closed interval [0, 2π] for the function f(x) = e^cos(x), you can follow these steps:
Step 1: Set up your graphing calculator
- Make sure your graphing calculator is turned on and ready.
- Press the "y=" key to enter the function.
Step 2: Enter the function into the calculator
- For the function f(x) = e^cos(x), enter "e^(cos(x))" into the calculator.
Step 3: Set the viewing window
- Press the "window" key to set up the viewing window for the graph.
- Since we are interested in the closed interval [0, 2π], you can set the x-minimum as 0 and the x-maximum as 2π.
- Set the y-minimum and y-maximum values to cover the range of the graph.
- You can start with the default values and adjust them later if needed.
Step 4: Graph the function
- Press the "graph" key to display the graph of the function on the calculator screen.
Step 5: Observe the graph
- Look at the graph and locate the points where the graph of f'(x) crosses the x-axis.
- These points represent the zeros of f'(x) on the interval [0, 2π].
Step 6: Count the number of zeros
- Count the number of times the graph of f'(x) crosses the x-axis on the interval [0, 2π].
Step 7: Determine the answer
- Based on the count from Step 6, determine the number of zeros for f'(x) on the closed interval [0, 2π].
- Choose the corresponding option (A, B, C, or D) from the given choices.
So, follow the steps outlined above with your graphing calculator to determine the number of zeros for f'(x) on the closed interval [0, 2π] for the function f(x) = e^cos(x).
f'(x) = -sinx e^cosx
since e^cosx is never zero, f' has zeroes where sinx has zeros.
I'm sure you remember enough trig to figure that out.