Calculus II.pdf

Calculus II
Practice Problems 1: Answers


1. Solve for x:

a) 6x 362
x


Answer. Since 36 62 , the equation becomes 6 x 62 2
x  , so we must have x 2  2  x  which has the
solution x 4  3.

b) ln3 x 5

Answer. If we exponentiate both sides we get x 35 243.

c) ln2  x  1  ln2  x  1 ln2 8

Answer. Since the difference of logarithms is the logarithm of the quotient, we rewrite this as
x 1
ln2   ln2 8
x 1
which is, after exponentiating, the same as
x 1
8
x 1
which gives us x  1 8  x  1  , so that x 9  7.



2. Find the derivative of the given function:

a) y ln  ln x 

Answer. Use the chain rule:
dy 1 d 1
ln x
dx ln x dx x ln x
b) y log2  x2  1 

Answer. Remember that log2 A ln A  ln2, so y  ln  x 2  1   ln 2. Then, use the chain rule:
dy 1 2 x
2x
dx  ln 2   x2  1  ln 2 x2  1 
2
ex
c) y
x
Answer. Use the quotient rule carefully:

x  2xex   ex
2 2

2ex  x
2 ex
dy 2 2

dx x2 
ex  2  x
2 
2