Chapter 5: Blocks recap

Question 1

The area bounded by f and g between x = [a,b]

Question 2

S = solid between a and b

A = the cross-sectional area of S

Give the formula for the volume:

Question 3

The shell method formula for area between [a,b] for f(x)

Question 4

The washer method

Question 5

Average value of a function

Question 6

The Mean Value Theorem for Integrals

Answer 6

If f is continuous on [a,b], then there exists a number c in [a,b] such that:

Question 7

Define an inverse function

Answer 7

Let f be a one-to-one function with domain A and range B.

Then the inverse function f^-1 has domain B and range A and is defined by:

f^-1(y) = x <=> f(x) = y

for any y in B

Question 8

Cancellation equations

Answer 8

f^-1( f(x) ) = x

f( f^-1(x) ) = x

Question 9

Theorem on continuity of inverses

Answer 9

If f is a one-to-one continuous function defined on an interval, then its inverse function f^-1 is also continuous.

Question 10

Theorem on differentiability of inverses

Answer 10

If f is a one-to-one differentiable function with inverse function f^-1 and f’(f^-1(a)) != 0, then the inverse function is differentiable at a and:

Question 11

Definition of the Number e

Question 12

^d/_dx (e^u)

Answer 12

e^u . ^du/_dx

Question 13

Cancellation equations for log

Answer 13

• log_a(a^x) = x
• a^log_ax = x for every x>0

Question 14

3 Log properties:

Answer 14

• log_a(xy) = log_ax + log_ay
• log_a(** ^x/**_{<strong>y</strong>}) = log_ax - log_a​y
• log_a(x^r) = r.log_ax

Question 15

Change of base formula

Answer 15

log_ax = ^lnx/_lna

Question 16

Derivative of lnx

Answer 16

¹/_x

Question 17

Derivative of ln(u) where u is a function

Answer 17

**¹/_u **. ^du/_dx

Question 18

Derivative of [ln g(x)]

Answer 18

^g’(x)/_g(x)

Question 19

Derivative of ln |x|

Answer 19

¹/_x

Question 20

Integral of ¹/_x

Answer 20

ln |x| + C

Question 21

Integral of tanx

Answer 21

ln |sec x| + C

Question 22

Derivative of log_ax

Answer 22

¹/_x.lna

Question 23

Derivative of a^x

Answer 23

a^x ln a

Question 24

Integral of a^x

Answer 24

a^x/_lna + C a != 1

Question 25

3 Steps in Logarithmic Differentiation

Answer 25

1. Take natural logarithms of both sides of an equation y = f(x) and use the properties of logarithms to simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y’

Question 26

d/dx a^b

Answer 26

0

Question 27

d/dx [f(x)]^b

Answer 27

b[f(x)]^b-1 f’(x)

Question 28

d/dx a^g(x)

Answer 28

a^g(x) (ln a).g’(x)

Question 29

Method to solve d/dx [f(x)] ^g(x)

Answer 29

Use logarithmic differentiation