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Consider the R²-R function defined by f(x,y)=x²-2x-y and let c be the contour curve of f at level 0.


1.find a Cartesian equation for the tangent line L to C at the point (-1,3).


2.sketch the contour curve C together with the line L in R².show all intersections with the axes.


3.find an equation for the tangent plane V to the graph of f in R³ at the point (-1,3,0).

1.approximately to what year does this evidence date back?

2.Give detail of how counting was done in the example which you chose.

3.where in caps is this type of counting covered as a topic?


A delivery company needs the measurements of a rectangular box such that the length plus twice the width plus twice the heights should not exceed 150cm.what is the maximum volume of such a box?

Consider the R²-R function f defined by f(x,y)=x²+2y²-x²y.


Show that f has two saddle points

Prove that a ring with one consists only of zero if and only if 1 = 0


Consider the R²-R function f defined by f(x,y)=e^xIn(1+y).


Find the third order Taylor polynomial of f about the point (0,0)

A certain group of welfare recipients receives relief goods with a mean amount of Php 500.00 per week. A random sample of 75 recipients is survived and found that the mean amount of relief goods they received in a week is Php 600 and a standard decision of Php. 50.00. Test the claim at 1% level of significance is not Php 500.00 per week and assume that the population is approximately normally distributed. What type of test does the problem represent?


Find the four roots of z⁴=-1


A certain group of welfare recipients receives relief goods with a mean amount of Php 500.00 per week. A random sample of 75 recipients is survived and found that the mean amount of relief goods they received in a week is Php 600 and a standard decision of Php. 50.00. Test the claim at 1% level of significance is not Php 500.00 per week and assume that the population is approximately normally distributed. Which of the following is correct null and alternative hypothesis, in symbol?


Show whether or not p←→ q =(p→q) ^ (q→p)

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