00:01
This problem, let f be the implicit function, which is equal to negative cosine of pi x plus 5x squared y, plus 6 times e to the power of x z, plus 2y z equal to 22.
00:24
Let's start off by finding the gradient of f.
00:27
Remember that the gradient of f is partial f, partial x, comma, partial f, partial y, comma, partial f, partial s.
00:37
So in our case, the gradient of f is going to equal to pi times sine of pi x plus 10x, y, plus 6 z times e to the power of x, z, comma, and then with respect to y, we are going to have 5x squared plus 2z comma and with respect to z we are going to have 6x times e to the power of x z plus 2y let's find the implicit let's find the gradient of f at point p where p here is right over here on the right side p is the point with coordinates 1 -30 and now we are going to evaluate the components of our gradient at the point 1 -30 so sign of 0 sign of a pi is 0 and so we are going to have 10 times 3 that's 30 so we're going to have 30 and then comma for the next component z is equal to 0 so we're going to have 5 and for the last component, remember that z is equal to zero.
01:59
So we're going to have 6 plus 6, which is 12.
02:02
So the tangent plane, it's going to be 30 times x minus 1 plus 5 times y minus 3.
02:15
And then plus 12 times z minus 0 equal to 0...