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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.
- Definitions : Please state in your own words the following definitions: a) Limit of a function z^ f (^ x , y ) b) Continuity of a function z^ f (^ x , y ) c) partial derivative of a function f(x,y) d) directional derivative of a function f(x, y) in the direction of a unit vector u e) definition of a function being differentiable f) gradient g) total differential h) The double integral of f over the region R f x ydA R
i) Center of gravity j) Surface area
- Theorems: Describe, in your own words, the following: a) a theorem relating differentiability with continuity b) a theorem relating differentiability with partial derivatives c) a theorem stating criteria for a function to have relative extrema d) a result that classifies critical points into relative max., min., or saddle points e) the procedure to find relative extrema of a function f(x, y) f) the procedure to find absolute extrema of a function f(x, y) g) how to switch a double integral to polar coordinates h) a theorem that allows you to evaluate a double integral easily
- True/False questions:
a) If lim( x , y^ ) f ^ (( x 0 ,, 0^ y ))^ ^0 then lim^ xf ( 0^ x ,^0 )^ ^0
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
f x y dxdy
( ) ( )
m) If f(x,y) is continuous then ^
d c b a b a d c
f ( x , y ) dydx f ( x , y ) dxdy
n) If f(x,y) is continuous then
o)
^ d c b a b a d c
f ( x ) g ( y ) dydx f ( x ) dx g ( y ) dy
- Surfaces : Find the domain for the following functions a) f ( x , y ) 4 x^2 y^2 b) f^ x^ y xy
c) (^22)
x y f x y
d) (^22)
x y f x y
- Limits and Continuity : Determine the following limits as (x,y) -> (0,0) and state which function is continuous everywhere, if any. a) 1
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
- Picture : Match the following contour plots (level plots) to their corresponding surfaces. e) [1] [2]
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
R R
R R
R
. Suppose R1 is
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?
- Directional Derivatives : a) Find the directional derivative of f(x, y) = xy exy^ at (-2, 0) in the direction of a vector u, where u makes an angle of Pi/4 with the x-axis. b) Suppose f^ (^ x , y^ )^ x^2^ ey. Find the maximum value of the directional derivative at (-2, 0) and compute a unit vector in that direction. c) For more examples, please see homework assignments
- Max/Min Problems: Compute the extrema as indicated a) f ( x , y ) 3 x 2 xy y 8 y 2 2 . Find relative extreme and saddle point(s), if any. b) f^ (^ x , y )^4 xy x^4 y^4. Find relative extrema and saddle point(s), if any c) Let f^ (^ x , y )^3 xy ^6 x ^3 y ^7. Find absolute maximum and minimum inside the triangular region spanned by the points (0,0), (3, 0), and (0, 5). i) For more examples, please see homework assignments
- Evaluate the following integrals: a) (^) 1 0 2 0 xy^2 dxdy b) (^) ^ 2 0 2 2 x 2 ydydx x x c) x y dydx x 3 3 9 0 2 2 2 d) (^) a b c x y z dxdydz 0 0 0 2 2 2
e)
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
- The pictures below show to different ways that a region R in the plane can be covered. Which picture corresponds to the integral f x ydxdy R
13. Suppose you want to evaluate
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.
- Use a multiple integral and a convenient coordinate system to find the volume of the solid:
a) bounded by z^ ^ x^2 y ^4 , z 0 , y^ ^0 , x 0 , and x 4
b) bounded by x^2
z e and the planes^ y^ ^0 ,^ y^ x^ , and^ x ^1
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
2 2 where R is a triangle bounded by^ y^ x^ ,^ y^ ^2 x ,^ x ^2
e) bounded by the paraboloid z 4 x^2 2 y^2 and the xy plane
- Find the following surface areas:
a) of the plane z^ ^2 x y above the rectangle 0 x 2 and 0 y ^3
b) of the cylinder z^ ^9 y^2 above the triangle bounded by y^ x^ , y^ ^ x , and y ^3
c) of the surface z 16 x^2 y^2 above the circle x^2 y^2 9
- Prove the following facts: a) A function f is said to satisfy the Laplace equation if 2 0 2 2 2
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
