 Suppose a curve x, y is expressed in parametric form with x and y both being functions of t. Then everything we know about derivatives can be used to find information about the curve. Now for this, it's helpful to remember one important thing about the derivative. The derivative is the rate of change of the function with respect to the variable. And in particular, the derivative will tell us when a function is increasing or decreasing, and when a function reaches an extreme value. For example, let's describe the behavior of the curve x of t, y of t, where x of t are given by these functions. So if you want to talk about behavior, it helps to know where you are. So first, we might find where we are at t equal to zero. And so we find... And so the curve passes through the point zero, zero. Now if we want to describe the behavior of the curve, we want to talk about the derivative. So we take the derivative of our functions x of t and y of t and find. And at t equal to zero, those derivatives will be... And so the thing to remember about the derivative, if the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing. So since x prime is equal to zero, then x is not changing. It's not increasing. It's not decreasing. Since y prime is zero is equal to one, then y is actually increasing at t equal to zero. So let's picture this. At t equal zero, the curve goes through the point zero, zero. And as it's going through that point, the x coordinate is not changing. We're not increasing, so we're not going to the right. We're not decreasing, so we're not going to the left. On the other hand, the y coordinate is increasing. And so that means the curve passes through this point going vertically upward. Now we can describe more complicated curves. Suppose a particle moves along some path given by these parametric equations. And maybe we want to know for what values of t will the particle be moving towards the upper left. So let's think about that. If the particle is moving to the upper left, that's this way, then its x coordinate is actually going to be decreasing. So we want x prime to be negative. Its y coordinate, on the other hand, will be increasing, and so we want y prime to be positive. So we find those derivatives. And since we're interested in knowing whether the derivative is positive or negative, we want to find the critical values. And since we have two derivatives, we'll actually have two sets of critical values. For x prime, our critical value will be, and for y prime, we'll get a critical value of, and now we want to find the sine of x prime and y prime. So we'll graph our critical values and determine the sine of x prime and y prime. So we'll graph. So the critical values partition the number line into three intervals, and so let's find the sines of x prime and y prime in each of these intervals. In the first interval, x prime is negative, and y prime is also negative. In this middle interval, x prime is negative, and y prime is positive. And in this last interval, x prime and y prime are both positive. Now remember we want to find where x prime is negative and y prime is positive, and so that's going to occur in this middle interval from negative 4 to 2. How about extreme values? For example, let's find the leftmost point on the graph of the curve with the same parametric equations. So let's think about that. If we're at a point on the curve and we can move to the left, we're not at the leftmost point. So let's take a look at the derivatives. Fortunately, we've already found those derivatives. And we also found the critical values, which allowed us to find the sines of x prime and y prime. And before t equals 2, x prime of t is less than zero, so we're moving to the left. And after t equals 2, x prime of t is greater than zero, so we're moving to the right. So if we're moving left until we hit t equals 2 and then move right, that means that at t equals 2, we are as far left as we can go. And so the leftmost point occurs at t equal to 2. And we do want the coordinates of that point, so at t equals 2, we find our leftmost point.