Use Trapezoidal rule to approximate the integral ∫12xdx, with n=5.[5]
3.
Solve ∫03∫12x2ydxdy[5]
Applications of Derivatives
1.
Estimate the area between the curve y=x2 and the lines x=0 and x=1, using rectangle method, with four sub intervals.A particle moves a line so that its velocity v at time t is
(1) Find the displacement of the particle during the time period 1≤t≤4
(2) Find the distance travelled during this time period.[10+0]
2.
Dry air is moving upward. If the ground temperature is 20∘ and the temperature at a height of 2km is 10∘, express the temperature T in ∘C as a function of the height h(in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km?[5]
3.
Find the equation of the tangent at (1,3) to the curve y=x2+1.[5]
4.
State Rolle's theorem and verify the theorem for f(x)=x2−9, x∈[−3,3].[5]
Derivatives
1.
Find the derivative of r(t)=t2i−te−tj+sin(2t)k and find the unit tangent vector at t=0.[5]
Function of One Variable
1.
If a function is defined by f(x)={1+x,x2,x≤−1x>−1, evaluate f(−3), f(−1) and f(0) and sketch the graph.Prove that limx→0x∣x∣ does not exist.[10+0]
2.
Sketch the curve y=x2+1 with the guidelines of sketching.If z=xy2+y3, x=sint, y=cost, find dtdz at t=0.[10+0]
3.
Starting with x1=1, find the third approximate x3 to the root of the equation x3−x−5=0.[5]
Infinite Sequence and Series
1.
What is sequence? Is the sequence an=5+nn convergent?[5]
Ordinary Differential Equations
1.
Define initial value problem. Solve: y′′+4y′−6y=0, y(0)=1, y′(0)=0Find the Taylor's series expansion for cosx at x=0.[10+0]
Partial Derivatives and Multiple Integrals
1.
Find the partial derivative fxx and fyy of f(x,y)=x2+x3y2−y2+xy at (1,2).[5]
Plane and Space Vectors
1.
Find the angle between the vectors a=(2,2,−1) and b=(1,3,2).[5]