A smooth curve has the equation 𝑦 equals 𝑓 of 𝑥. The slope of the curve at each 𝑥 is equal to 𝑥 cubed. If the curve goes through the point one, negative one, which of the following is its equation? a) 𝑦 equals four 𝑥 to the fourth power minus five, b) 𝑦 equals 𝑥 to the fourth over four plus five, c) 𝑦 equals 𝑥 to the fourth power plus five over four, or d) 𝑦 equals 𝑥 to the fourth power minus five over four.
We know that our curve is our function of 𝑥. We’re also given the information of a slope at each 𝑥 that’s equal to 𝑥 cubed. We can then say that the first derivative of whatever our function is is equal to 𝑥 cubed. We can also say that our function of 𝑥 could be found by integrating 𝑥 cubed with respect to 𝑥. To integrate 𝑥 cubed, it becomes 𝑥 to the fourth power over four plus some constant value of 𝑐.
We now have a general form of our equation. And we’re interested in figuring out what this constant is. Because we know a point that falls on our curve, we can plug in one for our 𝑥-value. And we know that 𝑓 of one equals negative one. That means that negative one is equal to one-fourth plus 𝑐. So we subtract one-fourth from both sides. And we find that 𝑐 equals negative five over four. So we plug that in for 𝑐 like this, which we simplify to say the function of 𝑥 𝑓 of 𝑥 equals 𝑥 to the fourth power minus five over four. And we know our curve 𝑦 equals 𝑓 of 𝑥. So we say 𝑦 equals 𝑥 to the fourth power minus five over four. And that’s option d).
There is a second strategy for solving this problem. Because we were given the point that our curve must go through, one, negative one, we could check the results for the point one, negative one for all four options. When we solve for option a), we get negative one equals four minus five. Negative one does equal negative one. So we couldn’t eliminate option a).
Option b), negative one equals one to the fourth power over four plus five which will be negative one equals one-fourth plus five. Negative one is not equal to five and one-fourth or 5.25. And we can eliminate option b). Option c), negative one equals one plus five over four. Six over four is not equal to negative one. And we can eliminate option c).
Using this method, choosing between a) and d), you can then calculate the derivative of a) and d). The derivative of a) would be 16𝑥 cubed. And the derivative for option d) would be equal to four times 𝑥 cubed over four which reduces to 𝑥 cubed, which was the original slope of the curve at each 𝑥 we were given. This elimination method works because we were given four different options.
Both methods show us that the equation for this curve is 𝑦 equals 𝑥 to the fourth power minus five over four.