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السلام عليكم يا اخوان انا عجزت بمعنى الكلمة وما خليت احد انه يساعدني ومافي فايدة ابي احد يساعدني باسئلة الرياضيات لا هنتم

Question-1: (15 marks)
Let f(x) be a polynomial of degree 3 which has a critical point at and an inflexion point at .
(6 points) Determine f(x) .








(2 points) Find another critical point of f(x) (if any).








( 4 points ) Determine the intervals where f(x) is increasing and where it is
decreasing .











( 3 points ) Use the first derivative test to find the local extrema of f(x)
















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Question-2: (15 marks)
Consider the function
( 5 points ) Find the equation of the line normal to the graph of f(x) when .














( 5 points ) Find the x coordinates of the points on the graph of f(x) where the tangent to the graph is horizontal.




















( 5 points ) Find the absolute extrema of the function on the interval [-2,2] .
























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Question-3: (15 marks) Consider the two parabolas : and
( 4 points) Find the points of intersection of the parabolas and decide which one is greater than the other between the intersection points.














( 4 points ) Compute the area enclosed by the two parabolas.












( 3 points ) Use Mathcad to draw the graphs of the parabolas and find their intersection points graphically .










( 4 points ) Solve the differential equation
.
















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Question-4: (15 marks)
(a) (4 points) Find antiderivatives for the following functions :
(i) (ii)










(b) Evaluate the following integrals over the given intervals:
(i) (2 points) over [1,3]








(ii) (3 points) cosh 3x-sinh4x over [0, ln2]










(c) A particle is moving from rest with an acceleration
(i) (3 points) Find the velocity of the particle after 25 seconds .












(ii)(3 points) Find the total distance travelled by the particle in the first 16 seconds.



تسون خير والله عجزت وانا ادور