By OpenStax

Calculus is designed for the common - or three-semester common calculus direction, incorporating cutting edge beneficial properties to augment scholar studying. The booklet publications scholars during the middle strategies of calculus and is helping them know how these options practice to their lives and the realm round them. as a result finished nature of the fabric, we're delivering the publication in 3 volumes for flexibility and potency. quantity three covers parametric equations and polar coordinates, vectors, capabilities of a number of variables, a number of integration, and second-order differential equations.

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**Example text**

B) The new setup, now that the ant has jumped onto the train wheel. (c) The ant travels along a prolate cycloid. 4. Using the same approach you used in parts 1– 3, find the parametric equations for the path of motion of the ant. 5. What do you notice about your answer to part 3 and your answer to part 4? Notice that the ant is actually traveling backward at times (the “loops” in the graph), even though the train continues to move forward. He is probably going to be really dizzy by the time he gets home!

This gives x′ (t) = 2 and y′ (t) = 3. Notice dx 2 dy dy/dt 3 = = . This is no coincidence, as outlined in the following theorem. 1: Derivative of Parametric Equations Consider the plane curve defined by the parametric equations x = x(t) and y = y(t). Suppose that x′ (t) and y′ (t) exist, and assume that x′ (t) ≠ 0. Then the derivative dy is given by dx dy dy/dt y′ (t) = = . 1) Proof This theorem can be proven using the Chain Rule. In particular, assume that the parameter t can be eliminated, yielding a differentiable function y = F(x).

The point (3, 2) is a relative minimum and the point (−1, 6) is a relative maximum, as seen in the following graph. 18 Graph of the curve described by parametric equations in part b. c. 1, first calculate x′ (t) and y′(t): x′ (t) = −5 sin t y′ (t) = 5 cos t. Next substitute these into the equation: dy dy/dt = dx dx/dt dy = 5 cos t dx −5 sin t dy = −cot t. dx cos t = 0 and is undefined when sin t = 0. This gives π 3π t = 0, , π, , and 2π as critical points for t. 19). On the left and right edges of the circle, the derivative is undefined, and on the top and bottom, the derivative equals zero.