Curve sketching and integration

Please see question attached. I require full detailed, step by step solutions to each section of this question.

Curve sketching and integration

Please see attachment for question. I require full, detailed, step by step workings for all sections of this problem Thanks

Correlation

1. (a) Show that if Z1 and Z2 are independent standard normal random variables, then for all ρ (correlation), Z1 and ρZ1+sqrt(1-ρ2)*Z2 are standard normal with correlation ρ. (b) Show that for all ρ and v, T1 = (Z1)/sqrt(X/v)and T2 = (Z2)/sqrt(X/v) have correlation ρ, where Z1and Z2 are standa ...continues

Tangents & Normals

i) Find the equation of the tangent to y=x(1-x) at x=1 ii) Find the equation of the normal to y=x(1-x) at x=1 iii) Find the equations of the tangents to y=x(1-x) that pass through (-1, 1/4)

Finding tangent through a certain point (involves drawing a diagram)

A portion of a river has the shape of the equation y=1-x^2/4, where distances are measured in tens of kilometres, and the positive y-axis represents due north. the town of Coopers Crossing is situated on the river at its most northerly point. The town of Black Stump is 10 kilometres due south of Coopers Crossing. the town of And ...continues

Implicit differentiation

If x^2+y^3=y+y^4, then find dy/dx

Radius of Convergence

Find the radius of convergence of the series in the attached file 'Series.doc'.

Definition of "Discontinuous function" Conditions for "Discontinuity" Examples on each type of discontinuity

Difine a discontinuous function and state the conditions for discontinuity. Find whether the following functions are discontinuous: f(x)=1/x and f(x)=(x)^(1/2) Solve the following:(involves jump discontnuity) A tomato wholesaler finds that the price of newly harvested tomatoes is $16 per pound if he purchases fewer tha ...continues

Discontinuity of an oscillatory function

Let f(x)=sin(1/x) if x is not equal to 0 =0 if x is equal to 1 Check whether f is continuous at 0(zero)?

Intermediate value theorem: Definition, Illustration by Graph and its application

Define Intermediate-Value theorem Using this theorem, show that there is a root of P(x)=x^3+x^2+x-1 in the closed interval [0,1].