The slope at the point 𝑥, 𝑦 on a graph of a function is d𝑦 by d𝑥 equals negative four 𝜋 sin of 𝜋𝑥 plus five 𝜋 cos of 𝜋𝑥. Find the equation of the curve if it contains the point one, two.
In this question, we’ve been given information about the derivative of our original function. Now, we’re looking to find the equation of the curve, given some information about a point that it passes through. In other words, we need to establish an equation for 𝑦 in terms of 𝑥. And so we begin by recalling that the fundamental theorem of calculus tells us that integration and differentiation are essentially reverse processes. And so we’ll find an equation for 𝑦 by integrating our expression for d𝑦 by d𝑥 with respect to 𝑥.
In this case, 𝑦 is then the indefinite integral of negative four 𝜋 sin of 𝜋𝑥 plus five 𝜋 cos of 𝜋𝑥 with respect to 𝑥. And actually, we can quote the general result for the integral of the sine and cosine functions. The integral of sin of 𝑎𝑥 for some real constants 𝑎 is negative one over 𝑎 cos of 𝑎𝑥 plus a constant of integration 𝑐. Similarly, integrating cos of 𝑎𝑥 with respect to 𝑥 is one over 𝑎 times sin of 𝑎𝑥 plus 𝑐. Remember, we can integrate term by term. So we begin by integrating negative four 𝜋 of sin 𝜋𝑥 with respect to 𝑥. It’s negative four 𝜋 times negative one over 𝜋 cos of 𝜋𝑥, which is four cos of 𝜋𝑥.
Then when we integrate five 𝜋 cos of 𝜋𝑥, we get five 𝜋 times one over 𝜋 sin of 𝜋𝑥, which is five sin of 𝜋𝑥. And since we’re performing an indefinite integral, it’s really important that we have that constant of integration 𝑐. So we’ve found that 𝑦 is equal to four cos of 𝜋𝑥 plus five sin of 𝜋𝑥 plus 𝑐. Now, we really need to find the value of 𝑐. So we go back to the question and the information that the curve passes through the point one, two. In other words, when 𝑥 is equal to one, 𝑦 is equal to two. And we substitute these values into our equation. We get two equals four cos of 𝜋 times one plus five sin of 𝜋 times one plus 𝑐.
Now, we know that five sin of 𝜋 is zero. Similarly, cos of 𝜋 is negative one. So four cos of 𝜋 is negative four. And our equation becomes two equals negative four plus 𝑐, which we can solve for 𝑐 by adding four to both sides. So 𝑐 is equal to six. And we can, therefore, replace 𝑐 with six in our equation for 𝑦. That’s 𝑦 equals four cos of 𝜋𝑥 plus five sin of 𝜋𝑥 plus six. Or if we choose by convention to write summing first, we get 𝑦 equals five sin of 𝜋𝑥 plus four cos of 𝜋𝑥 plus six.