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Find the centroid of the area bounded by the parabola y=x2 and the line 2x+3.
Find the moment of inertia of the area of the loop y2=x2(1-x), with respect to y-axis
Find the area of the surface that is generated by revolving the portion of
the curve y = x
3 between x = 0 and x = 1 about the x −axis.
Integrate ln(2x+2) from -1<=x<=1
a) find and classify the critical points of the functions f(x) = 2x^3 + 3x^2 - 12 x +1 into maximum, minimum and inflection points as appreciate.
(b) The sum of two positive numbers is S. find the maximum value of their product.
Sketch a possible graph of a function f that satisfies the following conditions:
(i) f(0) =0, f(2) = 3, f(4) =6, f'(0) =f'(4) =0
(ii) f'(x) >0 for 0<x<4, f'(x) <0 for x<0 and for x>4
(iii) f"(x) >0 for x<2, f"(x) <0 for x>2
d/dx(x^2)
Let f: I → R, where I is an open interval containing the point c, and let k ∈ R. Prove the following.
(a) f is differentiable at c with f ′(c) = k iff limh→0 [ f (c + h) – f (c)]/h = k.
*(b) If f is differentiable at c with f ′(c) = k, then limh→ 0 [ f (c + h) – f (c – h)]/2h = k.
(c) If f is differentiable at c with f ′(c) = k, then lim n →∞ n[f (c + 1/n) – f (c)] = k.
(d) Find counterexamples to show that the converses of parts (b) and (c) are not true.
The book is Steven R. Lay, Analysis with an introduction to proof.
Find the equation of the tangent line at the given value of x on the curve.
y^3+xy^2-62=x+2y^2 ; x=2
