WEBVTT
00:00:02.490 --> 00:00:06.780
The point three, three lies on the curve π¦ equals π₯ [7π₯] squared plus ππ₯ plus π.
00:00:07.270 --> 00:00:12.060
If the slope of the tangent there is negative one, what are the values of the constants π and π?
00:00:13.260 --> 00:00:17.180
Now, in order to begin to solve this problem, what weβre gonna look at first here is this statement.
00:00:17.390 --> 00:00:21.300
The statement says that the slope of the tangent there is negative one.
00:00:21.700 --> 00:00:30.370
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.
00:00:31.200 --> 00:00:35.830
So therefore, the first thing weβre gonna do is actually find the slope function of our curve.
00:00:35.960 --> 00:00:37.910
And the way to do that is by differentiation.
00:00:39.300 --> 00:00:44.400
And just to remind us how weβre actually gonna differentiate a function, what we can do is remind ourselves with the general rule.
00:00:44.520 --> 00:00:54.640
And if we have the function in the form ππ₯ to the power of π, then weβre gonna get the first derivative β so ππ¦ ππ₯ β is gonna be equal to πππ₯ to the power of π minus one.
00:00:55.000 --> 00:01:02.760
So what that means is the coefficient multiplied by the exponent and then multiplied by π₯ to the power of and then you reduce the exponent by one.
00:01:02.960 --> 00:01:05.020
Okay, so now, weβve just reminded ourselves of that.
00:01:05.020 --> 00:01:07.160
Letβs get on and actually differentiate our function.
00:01:08.030 --> 00:01:20.920
So if we differentiate our function, we get 14π₯ plus π because seven π₯ squared if we differentiate that, itβs seven multiplied by two β so the coefficient multiplied by exponent β and then π₯ to the power of two minus one which is just π₯.
00:01:21.280 --> 00:01:23.540
So great, so now, we know the slope function.
00:01:25.060 --> 00:01:28.110
So we now take a look back at the question and look at the information we had again.
00:01:28.280 --> 00:01:32.030
And it said that our slope of the tangent at this point is gonna be negative one.
00:01:32.260 --> 00:01:36.340
So therefore, as we said before, the slope of our curve here is also gonna be negative one.
00:01:37.720 --> 00:01:42.210
So therefore, the next step is going to actually to substitute ππ¦ ππ₯ for negative one.
00:01:43.460 --> 00:01:46.770
So thatβs gonna give us negative one is equal to 14π₯ plus π.
00:01:48.160 --> 00:01:53.190
So now, we can have a look at another bit of information from the question and thatβs that the point weβre dealing with is three, three.
00:01:54.390 --> 00:01:58.120
So therefore, we can say that at this point π₯ is gonna be equal to three.
00:01:58.420 --> 00:02:00.210
So as we know, this is the point that weβre dealing with.
00:02:00.250 --> 00:02:03.850
We can actually substitute π₯ equals three into our slope function.
00:02:05.160 --> 00:02:10.140
So therefore, we have that negative one is equal to 14 multiplied by three plus π.
00:02:11.540 --> 00:02:17.310
So therefore, what we actually do is we can actually subtract 42 from each side because 14 multiplied by three gives us 42.
00:02:17.710 --> 00:02:21.250
And we finally arrive at negative 43 equals π.
00:02:21.570 --> 00:02:24.600
Okay, great, so weβve now found the value of our constant π.
00:02:26.070 --> 00:02:27.820
And now, weβre moving on to the next part of the question.
00:02:27.850 --> 00:02:30.040
And what we now need to do is actually find out constant π.
00:02:30.360 --> 00:02:31.260
But how weβre gonna do that?
00:02:32.690 --> 00:02:38.440
Well, we said earlier that at the point three, three, π₯ is equal to three because that helped us find π through the slope function.
00:02:38.870 --> 00:02:43.080
But what we can do now is we also know that it tells us that π¦ is equal to three at this point.
00:02:44.420 --> 00:02:58.020
So therefore, what we can actually do to find π is substitute π₯ equals three, π¦ equals three, and π equals negative 43 into our original function, which is π¦ is equal to seven π₯ squared plus ππ₯ plus π.
00:02:59.450 --> 00:03:06.770
And when we do this, weβre gonna get three is equal to seven multiplied by three squared plus negative 43 multiplied by three plus π.
00:03:06.770 --> 00:03:09.560
And thatβs because weβve actually substituted in our values for π₯, π¦, and π.
00:03:11.080 --> 00:03:15.170
So this is gonna give us three is equal to 63 minus 129 plus π.
00:03:16.400 --> 00:03:19.560
So then, weβre gonna have three is equal to negative 66 plus π.
00:03:21.020 --> 00:03:24.260
So therefore, we arrive at 69 being equal to π.
00:03:25.090 --> 00:03:40.110
So therefore, we can say that given that the point three, three lies on the curve π¦ equals seven π₯ squared plus ππ₯ plus π and the slope of the tangent there is negative one, then the value of the constants π and π are going to be negative 43 and 69, respectively.