I can answer questions from the standard four semester Calculus sequence. I am not prepared for questions on Tensor Calculus. Everything else is welcome. Derivatives, partial derivatives, ordinary differential equations, single and multiple integrals, change of variable, vector integration (Green`s Theorem, Stokes, and Gauss) and applications.
Ph.D. in Mathematics and many years teaching Calculus at state universities.
B.S. , M.S. , Ph.D.
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Let l be the length and w be the width of the pasture. The area is wl . There are 2l + 3w meters of fencing used , so 400 = 2l + 3w .Then l= 200 - (3/2)w So the area is wl = (w)(200 - (3/2)w) = 200w
Let t be the number of trees planted. The number of oranges per tree will be 400 - 20 (t - 16) = 720 - 20t So the total yield will be y = (t)(720-20t) y = 720t - 20t^2 y' = 720 - 40t
The absolute maximum occurs where the derivative is 0 or at one of the endpoints . You found the derivative correctly. Set it equal to zero. 0 = 1-1/x^2 1/x^2 = 1 x^2 = 1 So x=1 is the only value
Let the radius of the cylinder be r and the height be h. Volume of the cylinder is πhr^2 Volume of the two hemispheres is (4/3)πr^3 So the volume of the solid is 6 = πhr^2 + (4/3)π
Revenue from selling x items : R(x) = (x)(200-.01x) Total cost of making x items : C(x) = 50x + 20,000 Profit from selling x items is revenue minus cost P(x) = 200x - .01x^2 - 50x -