

20 marks) 1. Estimate the instantaneous rate of change of f(x) - x at x =...
Calculate the work done by the vector field F(x,y)=4xy,
2x2
along a smooth, simple curve from point (3, −1) to point (4, 2)
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(a) For what value of x does f(x) = g(x)? (b) For which values of x is f(x) >g(x)? For every x in the interval f(x)>g(x). (Type your answer in interval notation.) In parts (a) and (b), use the given figure. (a) Solve the equation: f(x) = g(x). (b) Solve the inequality f(x) > g(x). у y = g(x) y = f(xN (-9,7)
* For the function f(x) = (3x + 5 if x 20 6 if x < 0. find f(-10)
f(x,y)= 0 1. (15 marks) Suppose X and Y are jointly continuous random variables with probability density function 12, 0<x<1, 0<y<0.5 else a) (5 marks) Find P(X - Y <0.25). b) (5 marks) Find P(XY <0.30). c) (5 marks) Find V (2x - 5Y+30).
Solve the inequality f(x) <0, where f(x) = - x2(x + 4), by using the graph of the The solution set for f(x) <0 is. (Type your answer in interval notation.) function. Ay 4- 2- х 500 -8 -6 -4 -2 2 4. 6 -8- -104 -12-
Given that 0<x< 2π, solve the followi ing inequalities. (2) 2sin2x <3cosx (3) 2cosx3 sinx+10
Question 9 Evaluate the integral f(x) dx where 203 f(x) = for x <1 for x > 1 6 7 4 5 3 O2 11 2
(9) Stokes' Theorem for Work in Space F(x, y, z) =< P,Q,R >=<-y+z, x - 2,x - y > S:z = 4 - x2 - y2 and z>0 (9a) Evaluate W= $ Pdx + Qdy + Rdz с (9) Stokes' Theorem for Work in Space F(x, y, z) =< P,Q,R>=<-y+z, x - 2, x - y > S:z = 4 - x2 - y2 and z 20 (9b) Verify Stokes' Theorem.
f(x,y)=0 2. (20 marks) Suppose X and Y are jointly continuous random variables with probability density function fc, 0<x<1, 0<y<1, x + y>1 else a) (2.5 marks) Find the constant, c, so that this is valid joint density function. b) (5 marks) Find P(Y > 2X). c) (5 marks) Find P(X>0.5 Y = 0.75). d) (5 marks) Find P(X>0.5 Y <0.75). e) (2.5 marks) Are X and Y independent? Justify your answer citing an appropriate theorem.
Let f(x)=
if
,
if
if
a) What is the fomain of f(x)? Write in interval notation.
b) Determine the y-intercept of the function, if any. Make sure
to justify your answer.
c) Determine the x-intercepts of the function, if any. Justify
your answer.
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