Rotate the "xyz 3D-space" to view the yz-plane (so looking down the x-axis).
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The LINE in the YZ-plane through the purple point (y,z)=(2,2) and having slope m=-1
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has the equation (z-2) = (-1) (y-2), which simplifies
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to z = 4 - y.
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Let stick the line z= 4-y into the plane x=1 ... below, think of the "t" as our "y"
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Expression 20: left parenthesis, 1 , "t" , 4 minus "t" , right parenthesis1,t,4−t
domain t Minimum: negative 5−5
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 55
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Lessons learned.
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The partial derivative of z=f(x,y) w.r.t. y [so think of x as fixed], evaluated at the point (x_0, y_0), is the slope of the tangent line in the plane x=x_0 to the curve which is the intersection of graph of z=f(x,y) and the plane x=x_0.
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See def. of Partial Derivative Remark part 1.
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Set to TRANSLUCENT surface (wrench icon).
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Rotate to our usual 3D view.
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Another way to view m= -1
Hide this folder from students.
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Similarly.
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The partial derivative of z=f(x,y) w.r.t. x [so think of y as fixed], evaluated at the point (x_0, y_0), is the slope of the tangent line in the plane y=y_0 to the curve which is the intersection of graph of z=f(x,y) and the plane y=y_0.
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All Done .
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Ignore this set-up folder (colors and dotted lines. )