The slope at the point 𝑥, 𝑦 on the graph of a function is five 𝑥 minus two over 𝑥. Find the equation of the curve if it contains the point 𝑒, five 𝑒 plus three.
To answer this question, it’s first useful to notice that we’ve been given an expression for the slope of our function. Now, we know that the slope at any point is found by differentiating the expression for the original function with respect to 𝑥. So if we’re given a curve 𝑦 is equal to some function of 𝑥, then the derivative d𝑦 by d𝑥 is its slope; it’s a given point. So here, d𝑦 by d𝑥 is equal to five 𝑥 minus two over 𝑥.
Now, the fundamental theorem of calculus tells us that differentiation and integration are essentially opposite operations. And so if we integrate our expression for d𝑦 by d𝑥 with respect to 𝑥, we’ll get our equation for 𝑦. Now, in fact, this will be a general equation. We’ll need to use the fact that our curve passes through the point 𝑒, five 𝑒 plus three to find a particular equation. So let’s begin by integrating d𝑦 by d𝑥; that’s five 𝑥 minus two over 𝑥 with respect to 𝑥. Now, before we perform any integration, we should simplify our integrand. We’ll simplify by dividing each term in the numerator by 𝑥. So 𝑦 is equal to the indefinite integral of five minus two over 𝑥 with respect to 𝑥.
Now, let’s integrate term by term. The integral of five is five 𝑥. Then, the integral of one over 𝑥 is the natural log of the absolute value of 𝑥. So the integral of negative two over 𝑥 is negative two times the natural log of the absolute value of 𝑥. Now, of course, we’re working with an indefinite integral, so we need that constant of integration. So we find 𝑦 is equal to five 𝑥 minus two times the natural log of the absolute value of 𝑥 plus 𝐶.
But of course, to find the equation of our curve, we need to work out what 𝐶 is. Well, we were told that our curve passes through the point 𝑒, five 𝑒 plus three. In other words, when 𝑥 is equal to 𝑒, 𝑦 is equal to five 𝑒 plus three. So let’s substitute these values in. We get five 𝑒 plus three equals five 𝑒 minus two times the natural log of the absolute value of 𝑒 plus 𝐶. Now, we know that 𝑒 is itself a positive number, so we don’t need these absolute value signs. We also know that the natural log of 𝑒 is simply one. So negative two times the natural log of 𝑒 is negative two. And our equation is now five 𝑒 plus three equals five 𝑒 minus two plus 𝐶. We can subtract five 𝑒 from both sides. Then we add two. And we find 𝐶 is equal to five.
And so we found the equation of our curve given information about its slope and a point that it passes through. It’s 𝑦 equals five 𝑥 minus two times the natural log of the absolute value of 𝑥 plus five.