Integration and Differentiation Problems, Exams of Calculus

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Question 1: (15)

Evaluate the integral ∫ −

1

0

( ln( x )) dx

Question 2: (15Points +5Points) The continuous function f ( x )has horizontal tangents at x =− 2 , x = 1 and x = 3. It has a

vertical tangent at x = 0 , and no tangent at x = 2. It is concave up for − 3 < x < 0 and 5 < x < ∞. It is concave down for 0 < x < 2 and 2 < x < 5. a) Identify all critical points, maxima, minima, and inflection points of f ( x ). Justify your

answers. b) In which intervals is f ( x )decreasing, in which intervals is it increasing?

Question 4: ( 10 Points + 5 Points):

Show that e^ dx xe e c

− x = − − x − − x +

∫ 2 2 by

(a) integrating the left side using a substitution and/or Integration by Parts. (b) using any other method

Question 5: (10 Points) Find an integral expression for the volume of the solid obtained by rotating the triangle enclosed by the lines y = − x + 3 , y = 2 , and the y-axis around

a) the y-axis b) the line y = 1

Do NOT evaluate the integrals!

Trigonometry Part: (10 Points + 10 Points) 1) Find all x ∈[ −π , 2 π]so that cos( 2 ( x − π/ 3 ))=− 0. 5

2) Show that

a) cos( 3 x ) = 4 cos^3 ( x )− 3 cos( x )

b) cos 2 (tan^1 ( )) tan^1 ( x ) dx

d − (^) x = −