To find the slope of the tangent line for the curve r = -2 - 3cosθ when θ = π/4, we can follow these steps:
1. Differentiate the equation r = -2 - 3cosθ with respect to θ to find the derivative of r with respect to θ.
2. The derivative of r with respect to θ represents the rate of change of r with respect to θ, which gives us the slope of the tangent line at any point on the curve.
3. Using the chain rule, we can differentiate the equation r = -2 - 3cosθ.
- The derivative of -2 with respect to θ is 0, as it is a constant term.
- The derivative of -3cosθ with respect to θ can be found by applying the chain rule. The derivative of cosθ with respect to θ is -sinθ, and multiplying it by the coefficient -3 gives us -3sinθ.
4. Simplifying the derivative, we get dr/dθ = 0 - 3sinθ = -3sinθ.
5. Now, substitute the value θ = π/4 into the derivative -3sinθ to find the slope of the tangent line at θ = π/4.
- Plugging in θ = π/4 into -3sinθ, we get -3sin(π/4).
- The sine of π/4 is equal to 1/√2 or √2/2, so the slope of the tangent line at θ = π/4 is -3(√2/2).
- Simplifying further, the slope of the tangent line at θ = π/4 is -3√2/2.
Therefore, the slope of the tangent line for the curve r = -2 - 3cosθ when θ = π/4 is -3√2/2.