The point three, three lies on the curve 𝑦 equals
𝑥 [7𝑥] squared plus 𝑎𝑥 plus 𝑏. If the slope of the tangent there is negative one, what are the values of the constants 𝑎 and 𝑏?
Now, in order to begin to solve this problem, what we’re gonna look at first here is this statement. The statement says that the slope of the tangent there is negative one. So what we know about this is that if we have a slope that is equal to negative one of a tangent at a point on a curve, then the curve is gonna have the same slope at that point.
So therefore, the first thing we’re gonna do is actually find the slope function of our curve. And the way to do that is by differentiation. And just to remind us how we’re actually gonna differentiate a function, what we can do is remind ourselves with the general rule. And if we have the function in the form 𝑎𝑥 to the power of 𝑏, then we’re gonna get the first derivative — so 𝑑𝑦 𝑑𝑥 — is gonna be equal to 𝑎𝑏𝑥 to the power of 𝑏 minus one. So what that means is the coefficient multiplied by the exponent and then multiplied by 𝑥 to the power of and then you reduce the exponent by one.
Okay, so now, we’ve just reminded ourselves of that. Let’s get on and actually differentiate our function. So if we differentiate our function, we get 14𝑥 plus 𝑎 because seven 𝑥 squared if we differentiate that, it’s seven multiplied by two — so the coefficient multiplied by exponent — and then 𝑥 to the power of two minus one which is just 𝑥.
So great, so now, we know the slope function. So we now take a look back at the question and look at the information we had again. And it said that our slope of the tangent at this point is gonna be negative one. So therefore, as we said before, the slope of our curve here is also gonna be negative one. So therefore, the next step is going to actually to substitute 𝑑𝑦 𝑑𝑥 for negative one. So that’s gonna give us negative one is equal to 14𝑥 plus 𝑎.
So now, we can have a look at another bit of information from the question and that’s that the point we’re dealing with is three, three. So therefore, we can say that at this point 𝑥 is gonna be equal to three. So as we know, this is the point that we’re dealing with. We can actually substitute 𝑥 equals three into our slope function. So therefore, we have that negative one is equal to 14 multiplied by three plus 𝑎. So therefore, what we actually do is we can actually subtract 42 from each side because 14 multiplied by three gives us 42. And we finally arrive at negative 43 equals 𝑎.
Okay, great, so we’ve now found the value of our constant 𝑎. And now, we’re moving on to the next part of the question. And what we now need to do is actually find out constant 𝑏. But how we’re gonna do that? Well, we said earlier that at the point three, three, 𝑥 is equal to three because that helped us find 𝑎 through the slope function. But what we can do now is we also know that it tells us that 𝑦 is equal to three at this point.
So therefore, what we can actually do to find 𝑏 is substitute 𝑥 equals three, 𝑦 equals three, and 𝑎 equals negative 43 into our original function, which is 𝑦 is equal to seven 𝑥 squared plus 𝑎𝑥 plus 𝑏. And when we do this, we’re gonna get three is equal to seven multiplied by three squared plus negative 43 multiplied by three plus 𝑏. And that’s because we’ve actually substituted in our values for 𝑥, 𝑦, and 𝑎. So this is gonna give us three is equal to 63 minus 129 plus 𝑏. So then, we’re gonna have three is equal to negative 66 plus 𝑏. So therefore, we arrive at 69 being equal to 𝑏.
So therefore, we can say that given that the point three, three lies on the curve 𝑦 equals seven 𝑥 squared plus 𝑎𝑥 plus 𝑏 and the slope of the tangent there is negative one, then the value of the constants 𝑎 and 𝑏 are going to be negative 43 and 69, respectively.