Let's rotate the "desmos xyz 3D space" to view the yz-plane (so looking down the x-axis).
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Examine line in the YZ-plane through the purple point (2,2) and having slope equal to the partial derv. of f w.r.t. y evaluated at (1,2) ... i.e., m= -1
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Expression 18: "m" Subscript, "c" "h" "a" "n" "g" "e" "i" "n" "Z" , Baseline equals hsv left parenthesis, 180 , 0.8 , 1 , right parenthesismchangeinZ=hsv180,0.8,1
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Expression 19: segment left parenthesis, left parenthesis, 1 , 2 , 2 , right parenthesis , left parenthesis, 1 , 2 , 3 , right parenthesis , right parenthesissegment1,2,2,1,2,3
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Expression 20: "m" Subscript, "c" "h" "a" "n" "g" "e" "i" "n" "Y" , Baseline equals hsv left parenthesis, 310 , 0.8 , 1 , right parenthesismchangeinY=hsv310,0.8,1
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Expression 21: segment left parenthesis, left parenthesis, 1 , 2 , 3 , right parenthesis , left parenthesis, 1 , 1 , 3 , right parenthesis , right parenthesissegment1,2,3,1,1,3
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Expression 22: left parenthesis, 1 , 1 , 3 , right parenthesis1,1,3
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The equation line in the YZ-plane through the purple point (2,2) and having slope m= -1 is :
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z = 4 - y .
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Let stick the line z= 4-y into the plane x=1 ... in below expression, think of the "t" as our "y"
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Expression 26: left parenthesis, 1 , "t" , 4 minus "t" , right parenthesis1,t,4−t
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Now let's go back to the usual 3D view.
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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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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.
