math test

MATH 005B Exam #1 Name:______________________________

Show all of your work!! If the value of a definite integral happens to be zero, you

must explain and show exactly why.

1. (10 pts) A) State the definition of the natural logarithm function, Ln(x).

B) In the following picture, the shaded area under f(t) = 1/t is exactly 1/π. Find

the exact value of Y.

2. (8 pts) Let g(x) = √Ln(10 − x2)

A) Find the domain of g(x).

B) Find and simplify: g ‘(x).

3. (8 pts) Use the First Fundamental Theorem of Calculus to find dy/dx if 𝑦 =

∫ sin (𝑡)

𝑡 𝑑𝑡

𝑥2

0

4. (8 pts) Find 𝑑

𝑑𝑥 [𝑥 sin (𝑥)]

5. (6 pts) Use implicit differentiation to establish that 𝑑

𝑑𝑥 𝐴𝑟𝑐𝑐𝑜𝑠(𝑥) = −

1

√1−𝑥2

6. (10 pts) Solve the Initial Value Problem:

𝑦′ = 𝑥 3𝑒 𝑥 2 , y(0) = 1

7. (8 pts) Find the volume of the solid formed when the region enclosed by the x-

axis, the y-axis, and y = cos(x) from x = 0 to x = π/2 is rotated about the vertical

line: x = -1.

9. (32 pts) Solve the following – show all your steps! Leave your answers in exact

form:

A) ∫ 𝑒 𝑐𝑜𝑠 2(𝑥)sin (𝑥) 𝑑𝑥

5𝜋

4 3𝜋

4

B) ∫ 1

𝑥√𝐿𝑛(𝑥) 𝑑𝑥

𝑒 2

𝑒

C) ∫ 𝑒 −𝑠𝑡 cos (𝑡)𝑑𝑡

D) ∫ 𝑥 2𝐿𝑛(𝑥)𝑑𝑥

E) ∫ 𝑠𝑖𝑛2(5𝑥)𝑑𝑥

F) ∫ 𝑡𝑎𝑛5(𝑥)𝑑𝑥

10. (10 pts) Use L’Hospital’s Rule to find the following limits:

A) lim 𝑥→1

Ln(𝑥)

sin (𝜋𝑥) B) lim

𝑥→0+ cos (2𝑥)

1

𝑥2

Extra Credit: Show that:

Ln|sec(x) + tan(x)| = -Ln|sec(x) - tan(x)|