Calculus Answers

3. a) Define tangent and normal of a curve with figure. Also find the equation of tangent and normal of the ellipse (x ^ 2)/4 + (y ^ 2)/16 = 1 at the point (- 1, 3) .

b) Explain maximum and minimum value of a function with graphically. Evaluate maximum and minimum value of the function f(x) = x ^ 3 - 3x ^ 2 + 3x + 1

 Evaluate "\\intop"x²(1 + 2x³)³dx.

2. Evaluate "\\intop"xe^7x dx.

3. Find the volume of the solid of revolution when the curve y = 1 + x2 is revolved around the x-axis on [−2, 2].

Show that the curve with parametric equations

x = sin t and y = sin(t + sin t) for 0 ≤ t ≤ 2π

Solve the Cauchy Problem,

xu_x+(x+y)u_y=u+1 with u(x,y)=x² on y=0

2. a) Find the derivatives of the following functions with respect to x.

x ^ 3 + y ^ 3 = 3

y = (sin x) ^ tan x

b) Evaluate the 2 ^ (nd) order partial derivatives partial^ 2 u partial x^ 2 and partial^ 2 u partial y^ 2 if u=2x^ 3 +3x^ 2 y+xy^ +y^ .

Given that U

 is a function of x,y

 and z

 and A

 a vector field, prove that:

∇.(UA)=(∇U).A+U(∇.A).

a,Calculate the area under the curve 𝑦 = 𝑥3 + 4𝑥 + 1 from x=-3 to x=3. 5

b) Evaluate: ∫4 𝑥𝑒𝑥 dx.

a) Define tangent and normal of a curve with figure. Also find the equation of tangent and normal of the ellipse (x ^ 2)/4 + (y ^ 2)/16 = 1 at the point (- 1, 3) .

b) Explain maximum and minimum value of a function with graphically. Evaluate maximum and minimum value of the function f(x) = x ^ 3 - 3x ^ 2 + 3x + 1

Find the derivatives of the following functions with respect to x.

𝑥³ + 𝑦³ = 3                                                                                                                   

𝑦 = (sin 𝑥)^{𝑡𝑎𝑛𝑥}

A farmer wants to determine the dimensions of the largest rectangular area tgat can be inscribed in a right angled triagle field with a height h=4 meters and a hypotenuse of 5 meter

Find the dimensions of the rectangle with the maximun area