00:01
In this question, we are asked to find the points on the graph where the tangent line is horizontal and vertical, where the tangent line is horizontal and find the points where the tangent line is vertical.
00:13
To do that, we'll assume that y is a function of x, and y depends on x, and x is the independent variable.
00:23
And then differentiate the equation implicitly.
00:28
This means that whenever we differentiate an expression in the word, we need to multiply it by y prime and will differentiate x as usual the derivative of 64 x squared equals to 64 times 2 x the derivative of 49 y squared equals to 49 times 2y but since y depends on x we need to multiply it by y prime plus 896 minus 784 the derivative of y equals to 1 but since y depends on x we need to multiply it by y prime and the derivative of 3 ,136 equals to 0 because it's a constant and on the right hand side we are going to get 0.
01:21
Now we are going to keep terms with y prime on the left hand side we are going to get 98 y y prime minus 784 y prime equals to 100 128 x sorry negative negative 128 x minus 8996 and then after next let's factor out y prime on the left hand side we are going to get 98 y minus 784 times y prime equals to negative 128 x minus 828x minus 896 finally, let's divide both sides of the equation by the expression in front of y prime to get negative 128x minus 896 divided by 98y minus 784.
02:46
Now, the tangent line is horizontal if the numerator is zero, if y prime is zero.
03:01
And y prime is 0 if negative 128 x minus 896 is 0 therefore negative 128 x equals to 896 therefore x equals to negative 896 over 128 now this is this equals to 7 negative this is a value of x at which the tangent binds horizontal.
03:57
And to find the values of y, to find the values of y, we need to plug in the value of x we just found into the equation of the curve.
04:15
And after plugging in x equals negative 7 in the equation of the curve, we are going to get 64 times 49 plus 49 y squared minus 896 times 7.
04:32
Minus 784y plus 31 36 equals to 0 and this gives us a quadratic equation for finding y 49 y squared minus 784 y and now let's see what we're going to get after adding the constants 64 times 49 minus 8 96 times 7 plus 3136 times 7 plus 31 36 36 313 3 1 36 equals to 0.
05:25
Very nice.
05:27
So this is the equation for finding y.
05:31
Also, let's see if 784 is divisible by 49 equals to 16.
05:42
So the equation for finding y is y squared minus 16, y equals 0.
05:50
And therefore, y times y minus 16 is 0 and y equals 0.
06:05
Now recall that this corresponds to x equals negative 7.
06:10
In both cases, x equals to negative 7.
06:17
These are the points where the tangent line is horizontal.
06:26
Two points.
06:28
Negative 7 .0 and negative 716...