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This is a practice exam for a calculus iii course, covering various topics such as limits, continuity, differentiation, total differential, directional derivatives, and multiple integrals. It includes definitions, theorems, true/false questions, and problems to solve.
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Math 2411 – Calc III Practice Exam 2 This is a practice exam. The actual exam consists of questions of the type found in this practice exam, but will be shorter. If you have questions do not hesitate to send me email. Answers will be posted if possible – no guarantee.
i) Center of gravity j) Surface area
b) If lim^ yf ^ (^00 ,^ y )^ ^0 then lim( x , y^ ) f (( x 0 ,, 0^ y ))^ ^0 c) If f is continuous at (0,0), and f(0,0) = 10, then lim( x , y^ ) f (( x 0 ,, 0^ y ))^ ^10 d) If f(x, y) is continuous, it must be differentiable e) If f(x, y) is differentiable, it must be continuous f) If f(x, y) has partial derivatives fx and fy, then f must be differentiable g) If f(x, y) has partial derivatives fx and fy and both are continuous then f must be differentiable h) If f(x, y) is a function such that fxx, fyy, fxy, and fyx exist then fxy = fxy i) If f(x, y) is a function such that all second order partials exist and are continuous then fxy = fyx j) If f(x, y) is a function such that all second order partials exist and are continuous then fxx = fyy k) Every region in the plane can be classified as either a type-1 or type-2 region l) The volume under f(x,y), where a x b and g^ (^ x )^ y h ( x )is (^) b a g y g x
( ) ( )
m) If f(x,y) is continuous then ^
d c b a b a d c
n) If f(x,y) is continuous then
o)
^ d c b a b a d c
c) (^22)
x y f x y
d) (^22)
x y f x y
lim (^22) ( ,) ( 0 , 0 )
(^) x y xy x y b) (^) (,) ( 0 , 0 ) 2 2
lim x y xy x y
c) (^) (,^ lim) ( 0 , 0 ) 2 2 x y xy x y d) (^22) 2 (,) ( 0 , 0 ) lim x y x y x y e) (^22) 2 2 (,) ( 0 , 0 ) lim x y x y x y
b) Suppose the radius of a right cylinder is measured with a 2% error, while the height is measured with an error of 4%. What is the maximum relative error in V, the volume of that cylinder. c) The total resistance R of two resistances R1, R2 that are connected in parallel is 1 2 1 2
measured at 200 Ohm, with an error of 2%, and R2 is measured at 400 Ohm, with an error of 2% as well. What is the error in computing R from this data?
0 1 cos( x^2 ) dxdy y f) x y dA R
2 2 , where R is the part of the circle in the 1st^ quadrant g) (^) 0 / 2 0 sin( x )cos( y ) dydx h) For more examples, please see homework assignments
R f ( x , y ) dA where R is the region in the xy plane bounded by y (^0) , y 2 x 2 , and y^ x^. According to Fubini’s theorem you could use either the iterated integral (^) f ( x , y ) dxdy or f ( x , y )^ dydx to evaluate the double integral. Which version do you prefer? Explain.
b) bounded by x^2
c) bounded above by z 16 x^2 y^2 and bounded below by the circle x^2 y^2 4 d) evaluate (^) R x y y
e) bounded by the paraboloid z 4 x^2 2 y^2 and the xy plane
c) of the surface z 16 x^2 y^2 above the circle x^2 y^2 9
y f x f
. Show that the function f ( x , y )ln( x^2 y^2 )satisfies the Laplace equation. b) Two function u(x, y) and v(x, y) are said to satisfy the Cauchy-Riemann equations if (^) y v x u
and x v y u
. Show that the functions u ( x , y ) ex cos( y )and v ( x , y ) e x sin( y )satisfy the Cauchy-Riemann equations. c) Show that if u(x, y) and v(x, y) are functions such that all second-order partials are continuous and u and v satisfy the Cauchy-Riemann equation. Then both u and v also satisfy the Laplace equation. d) Use the definition of differentiability to show that f^ (^ x , y ) xy is differentiable. e) Let 0 , for( , ) ( 0 , 0 ) , for( , ) ( 0 , 0 ) ( , )^22 x y x y x y xy f x y Then show that f has partial derivatives at (0, 0) but f is not differentiable at (0, 0). f) Prove that the volume of a Sphere with radius R is 4/3 * Pi * r^3 g) Prove that the surface area of a Sphere with radius R is 4 * Pi * r^2