Search & Filtering
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
The area bounded by the curves y = x2 + 2 and y = 3x + 2 from intersection point to intersection point is rotated around x = 3.
Determine the formula for the the solid of revolution which is obtained from this rotation.
Find the Taylor polynomial of order 4 (at x = 0) for the function : f(x) = ex^2
(Sections 13.1–13.6) Let R be the region in the sketch below. MAT2615/001 (a) Describe the region R as a union of two Type 1 regions. (Use set–builder notation.) Hints: • Read the description of a Type 1 Region on p.18 of Guide 3 and study Fig. 13.9 carefully. • Shade the region R by means of vertical lines and highlight the curves which form lower and upper boundaries of R. Write the equations of these curves in the form y = g(x). • The lower boundary of R is formed by two different curves. (3) (b) Describe the region R as a union of two Type 2 regions. (Use set–builder notation.)
