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Solve the Cauchy Problem,

xux+(x+y)uy=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

1. a) Define differentiation and integration in calculus. Also write down the

differences between them. 

b) Write down some application of Calculus in CSE. 

c) Describe geometrical meaning of definite integral with figure.


for a particular function, dy/dx=8x-5. if it is known that when x= 2,y= 8,find y in terms of x


Let g(x) = {(ax^2-b) if x<2, (bx-a) if x>2}

Find a relationship between a and b (that is, solve for a in terms of b or vice versa) So that g(x) is continuous for all of x.