Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results.

f(θ) = 8 sin(θ) − θ, (0, 0)

f(θ)= 8sin(θ)-θ (0,0)

derivative of sin(x) is cos(x), so
f'(θ)=2cos(θ)-1
put the point 0 in for θ, it looks like this: f'(θ)=2cos(0)-1
cos(0)=1
so 8*1=8
8-1=7
The slope is 7

I realize this question is six years old, but hopefully this will help someone else that has this same problem.

Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your result . Function y=6x^_7,point(1,-1)

To find the slope of the graph of the function at the given point (0, 0), we need to take the derivative of the function with respect to θ.

Step 1: Find the derivative of f(θ):

f'(θ) = 8 cos(θ) - 1

Step 2: Evaluate the derivative at the given point (θ = 0):

f'(0) = 8 cos(0) - 1 = 8(1) - 1 = 8 - 1 = 7

Therefore, the slope of the graph of the function at the point (0, 0) is 7.

To confirm this result, you can use a graphing utility with a derivative feature.

To find the slope of the graph of the function at the given point, we need to find the derivative of the function. The derivative gives the rate of change of the function at any given point.

To find the derivative of the function f(θ) = 8 sin(θ) − θ, we will use the chain rule and the product rule.

Step 1: Find the derivative of the first term, which is 8 sin(θ).
The derivative of sin(θ) is cos(θ), so the derivative of 8 sin(θ) is 8 cos(θ).

Step 2: Find the derivative of the second term, which is -θ.
Since the derivative of a constant multiplied by a variable is just the constant, the derivative of -θ is -1.

Step 3: Combine the derivatives of the two terms. Since the two terms are added together in the original function, we can simply add their derivatives to find the derivative of the entire function.
So, the derivative of f(θ) = 8 sin(θ) − θ is: f'(θ) = 8 cos(θ) - 1.

Now, to confirm the result using the derivative feature of a graphing utility:
1. Graph the function f(θ) = 8 sin(θ) − θ on a graphing utility.
2. Use the derivative feature of the graphing utility to find the derivative of the function.
3. Evaluate the derivative at the given point (0, 0).
4. The value obtained from step 3 should match our earlier calculation of f'(θ) = 8 cos(θ) - 1.

By following these steps, you can find the slope of the graph of the function at the given point and confirm the result using a graphing utility.