67–72. Derivatives Consider the following parametric curves.
a. Determine dy/dx in terms of t and evaluate it at the given value of t.


x = cos t, y = 8 sin t; t = π/2

Verified step by step guidance

1. Recall that for parametric curves defined by \(x = x(t)\) and \(y = y(t)\), the derivative \(\frac{dy}{dx}\) can be found using the chain rule as \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
2. Given the parametric equations \(x = \cos t\) and \(y = 8 \sin t\), compute the derivatives with respect to \(t\): \(\frac{dx}{dt} = -\sin t\) and \(\frac{dy}{dt} = 8 \cos t\).
3. Substitute these derivatives into the formula for \(\frac{dy}{dx}\) to get \(\frac{dy}{dx} = \frac{8 \cos t}{-\sin t} = -8 \frac{\cos t}{\sin t}\).
4. Evaluate \(\frac{dy}{dx}\) at the given value \(t = \frac{\pi}{2}\) by substituting \(t\) into the expression \(-8 \frac{\cos t}{\sin t}\).
5. Simplify the expression after substitution to find the slope of the curve at \(t = \frac{\pi}{2}\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with these equations is essential for analyzing motion and curves in calculus.

Derivative of Parametric Curves (dy/dx)

For parametric curves, the derivative dy/dx is found by dividing the derivative of y with respect to t by the derivative of x with respect to t, i.e., (dy/dt) / (dx/dt). This method allows us to find the slope of the tangent line to the curve at a specific parameter value, even when y is not explicitly given as a function of x.

Evaluating Derivatives at Specific Parameter Values

After finding the general expression for dy/dx in terms of t, evaluating it at a specific value of t gives the slope of the tangent line at that point on the curve. This step involves substituting the given t-value into the derivative expressions and simplifying to find the numerical slope.