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Calculus Formulas for MATH 53 Midterms Question and answers
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Calculus Formulas for MATH 53 Midterms Question and answers Implicit function theorem Maxima and minima Definitions of local maxima and minima; finding critical points and determining their nature using the second derivative test; finding absolute minima and maxima. Constrained extrema and Lagrange multipliers To find maxima and minima of f under the constraint g = k solve ∇f = λ∇g, g = k. Double integrals
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Calculus 3 - Quadric Surfaces Question and answers
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Calculus 3 - Quadric Surfaces Question and answers z^2=x - Parabolic Cylinder HT: Parallel Lines VT: Parabola x^2+y^2=1 - Elliptic Cylinder HT: Ellipse VT: Rectangle (x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1 - Ellipsoid HT: Ellipse VT: Ellipse
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Calculus 3 Unit 1 Question and answers
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Calculus 3 Unit 1 Question and answers What sign does the first octant have in 3-space? Positive x,y,z coordinates What is the distance between two points on 3-space d = sqrt ((x2-x1)^2 + (y2-y1)^2 + (z2-x1)^2) Equation of sphere with center (h,k,l) and radius r r^2 = (x-h)^2 + (y-k)^2 + (z-l)^2 Cylindrical surface When a 2 dimensional curce is translated to a 3 dimensional surface Vector A quantity with magnitude and direction Zero vector Has a length of zero, no directi...
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Calculus 3 Question and answers
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January 2026

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Calculus 3 Question and answers Unit Vector - v÷(||v||) General Form Equation of a Sphere - Center =[(b1-a1)/2 , (b2-a2)/2 , (b3-a3)/2] Radius = √[(b1-C1)² + (b2-C2)² + (b3-C3)²] Equation = (x-C1)² + (y-C2)²+(z-C3)² Orthogonal Vectors - vectors u and v such that u dot v = 0 Area of Points - Area = 1/2 ||ABxAC|| Equation of a plane that passes through 3 points - PQxPR = <A, B, C>
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Calculus 3 Question and answers
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January 2026

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Calculus 3 Question and answers Distance Formula 3D - d = √[( x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] Equation for Sphere with Radius and Center - r^2 = (x - x0)^2 + (y-y0)^2+(z-z0)^2 Magnitude/Length Vector Equation - V = (a,b,c) Magnitude = sqrt (a^2+b^2+c^2) Special Vector(s). What do vectors normally have? - Components! i = (1,0,0), j = (0,1,0), k = (0,0,1) Unit Vector Equation - vector / mag of vector Dot Product (x,y,z) (x',y',z') - = (x)(x') + (y)(y...
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Calculus 3 Question and answers
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January 2026

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Calculus 3 Question and answers Distance formula - Equation of a Sphere - Midpoint formula - Arc Length - Adding and Subtracting vectors u + v u - v - <u1 + v1, u2 + v2, u3 +v3>
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Calculus 3 Question and answers
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Calculus 3 Question and answers Name three types of function domain restrictions: - Division by zero, even roots of negatives, log of non-positives What is the "footprint" of a surface? - The projection of the surface onto the xy-plane What are level curves? - Curves where f(x, y) = constant — a slice of the surface How are level curves different from traces? - Level curves fix the output (z); traces fix input (x or y) How are level curves like traces? - Both are 2D slices ...
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calculus 3 formulas Question and answers
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January 2026

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calculus 3 formulas Question and answers magnitude of a vector - unit vector - dot product - if the dot product is 0, the vectors are perpendicular angle between two vectors - vector projection of b onto a - scalar projection of b onto a - cross product - if cross product is 0, the vectors are parallel
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Calculus 3 FinalQuestion and answers
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January 2026

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Calculus 3 FinalQuestion and answers Scalar projection - (*a* x *b*)/ |*a*| Vector projection - ((*a* x *b*)/ |*a*|^2 ) x *a* Vector equation of a line through point (x, y, z) with direction vector *v* = <a, b, c> - *r*(t) = <x + at, y + bt, z + ct> Equation of a plan with point (xi, yi, zi) and normal vector *n* = <a, b, c> (made by *PQ* x *PR*) - a(x-xi) + b(y-yi) + c(z-zi) = 0 Unit tangent vector - *r*'(t) / |*r*'(t)| Unit normal vector - *T*'(t) / |...
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Calculus 3 FinalQuestion and answers
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January 2026

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Calculus 3 FinalQuestion and answers Scalar projection - (*a* x *b*)/ |*a*| Vector projection - ((*a* x *b*)/ |*a*|^2 ) x *a* Vector equation of a line through point (x, y, z) with direction vector *v* = <a, b, c> - *r*(t) = <x + at, y + bt, z + ct> Equation of a plan with point (xi, yi, zi) and normal vector *n* = <a, b, c> (made by *PQ* x *PR*) - a(x-xi) + b(y-yi) + c(z-zi) = 0 Unit tangent vector - *r*'(t) / |*r*'(t)| Unit normal vector - *T*'(t) / |...
DrJOHNJAMES
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