math final

Math 005B Final Exam Name:_____________________________________________________

Show all your work.

Part I: Find the antiderivatives – 10 pts each:

1. ∫ cot⁡(Ln(x))

3x dx

2. ∫sec(x)tan3(x)dx

3. ∫e2xsin⁡(3x)dx

4. ∫ x+2

x(x−4)2 dx

5. ∫ Ln(x3)

x2 dx

Part II: Evaluate the definite integrals (10 pts each)

6. ∫ x3−2x2

x2+1 dx

√2⁡⁡

0

7. ∫ 1

√𝑥2+4 dx

√3

1

8. ∫ esin 2(x)cos(x)dx

2π

3 π

3

– Remember you MUST show your work for any credit!

Part III Define the following – 2 pts each (You MUST use the definitions I

presented in class!):

9. Ln(x)

10. e

11. Arcsin(x)

12. sinh(x) and cosh(x)

Part IV: (8 pts total)

13. Put the Hyperbola in standard form, and find all of the features:

3x2 – 12x – y2 – 8y + 4 = 0 Center:_______________

Vertices:___________ and _____________

Foci: ______________ and _____________

Asymptotes:____________________

and ___________________________

Part V: 8 pts each:

14. Find the limit: lim x→0+

(cos(x) + 2x) 1

x

15. If 𝑟(𝜃) = 2sin(3𝜗)find the equation of the tangent line to the curve at ϑ= 𝜋

4

Part VI (8 pts):

16. Derive the formula for the derivative of the Arccosecant function, using implicit

differentiation.

Part VII Points as indicated:

17. (15 pts) Recall that the Maclaurin series for f(x) = 1

1+𝑥2 is ∑ (−1)𝑛𝑥2𝑛∞𝑛=0

A) Find the radius and interval of convergence – be sure to check the end points!

B) Use the above fact to find the Maclaurin series for g(x) = Arctan(x)

18. Decide whether the series ∑ (−1)nLn(n)

√n ∞ n=2 diverges, converges absolutely, or

converges conditionally. Show all your steps and indicate the tests you are using!

(10 pts)

19. (10 pts) Find the first three non-zero terms in the Maclaurin series for

f(x) = cos(3x) (centered at x = 0, of course.)

20. (5 pts) Find the sum of the series: ∑ 2𝑛+1

33𝑛−2 ∞ 𝑛=1