The minimum of this function is at the point (6,1) Changing variables to x' = x -6 and y' -1 results in f(x', y' ) = x'2 - 2 x' y' + 5 y'2 +24 It is possible to show that x'2 - 2 x' y' + 5 y'2 is always greater...

The minimum of this function is at the point (6,1) Changing variables to x' = x -6 and y' -1 results in f(x', y' ) = x'2 - 2 x' y' + 5 y'2 +24 It is possible to show that x'2 - 2 x' y' + 5 y'2 is always greater...

This is about laplace transform (answer)

Proving this presupposes that limt -> ∞ U(t) exists. Since it exits, call it Q Then consider the left hand side lims -> 0 s ∫ U(t) exp(- st) Interchanging the operation...

An analytic approach is to write the right hand side as: ( 1 + x)1/3 (1-x )1/3 Then define Q as Q = [ (1 -x )/(1 + x) ]1/3 After some rearrangement the equation becomes 1/Q ...

This problem can be solved with the given information. For part a) the external work = change of energy principle is useful Friction is doing negative work on the system. The change of energy of the system is Ef - Ei , where Ei is the gravitational...

The fact that the second ball has a 4 means that the first ball cannot have a 4. The probability that the first ball has a 1 is thus 1/(1414 -1) = 1/1413 = 1/471. Similar logic shows that the probability of the first ball having a 2 is also 1/471 and for a 3, again...

Use the triangle construction. Draw a right triangle with opposite side of length 15 and adjacent side of length 8. The hypotenuse is then sqrt( 152 + 82 ) = sqrt(289) = 17. The cosine is then adjacent over hypotenuse = 8/17...

Deriving functions Problem (answer)

The function f will be one to one onto if it is monotone increasing on R. f will be monotone increasing if its slope is everywhere greater than zero. All that is required for the slope to be everywhere greater than zero is for epsilon to be less than 1/M.

The closing speed is 60 + 80 = 140 mph the time to collision is thus 364/140 = 2.6 hrs or 2 hrs 36 min

calculus homework (answer)

Call 6 am t=0, then 9 am corresponds to t = 3 A straight line estimate of the rate as a function of t is: R(t) = 100 + 60 t. The volume if the water piped is the integral of R over t with the limits: ...

How do I solve this problem? (answer)

The formula for volume, V, is depth x width x length. The depth is x. With the squares cut out, the width is reduced to 10 - 2x and the length is reduced to 16- 2x. The working equation is then 137.5 = x (10- 2x) (16 - 2x) This...

The center is the midpoint of the diameter, so the coordinates of the center are (5, 11). The radius is the distance from the center to one of the diameter end points. The distance formula can be used to find that this is sqrt(10) Finally the equation of the...

The integral D is a concatenated integral with the outer integration (over x) from 0 to 1 and the inner integration (over y) being from sqrt(x) to 1. In graphical terms, the integration is over an area in the plane which is a subset of a 1 x 1 square with the lower left corner ...

A possible answer is f(x) = x2 /(x2 - x -2) This has the required asymptotes . The domain is not all real numbers. The values x = -1 and x = 2 must be excluded from the domain . However all other values...

The best approach seems to be to develop two expressions for the common tangent line: one for the parabola, the other for the circle. The tangent line will be of the form y = mx + b. Of course m and b must eventually be the same for both. For...

In the laser Raman effect, the wavelength of a small portion of the exciting optical radiation is shifted to a slightly longer wavelength. The longer wavelength is usually also in the optical part of the spectrum, As an example, consider using a green Argon ion...

The probability that at least one will fail is easy to work out. It is 1 - (.7)3 = .657 This the complement of the probability that all three pass ( prob = .73 ). Computation of the probability that exactly one will fail is harder...

Find multivariable limits (answer)

For the first one, x4 - y4 factors as (x2 - y2)(X2 + y2) the second factor cancels out leaving x2 + y2 which is zero in the limit indicated For the second one, change top polar coordinates: x = r cos(θ) ...

The midpoint of AB is MP = (0,0,-3) A unit vector from MP to C is u =(3,-1,2)/sqrt(14) In coordinate form, the parametric form of the line is C + t u or (3,-1,-1) + t (3,-1,2)/sqrt(14) The parametric...

The case b >1 is easiest to think about. With b >1 , the argument of function f changes more rapidly (as x increases) than would be the case with b =1. This means that anything that f is going to "do" happens more rapidly. More rapid...

The anti-derivative of 1/(6 x3) is -(1/12)/ x2 So the area under the curve is (1/12) ( 1 - 1/t2 ) So for t = 10 area = (1/12) .99 for t = 100 area = (1/12) (.9999) The entire area corresponds...