Calculus Chapter 6: Integral Calculus

Question 1

Disk method formula

Question 2

Washer method formula

Question 3

Shell Method formula

Question 4

Average value formula

Question 5

The Mean Value Theorem for integrals

Answer 5

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

Question 6

One-to-one function

Answer 6

A function that never takes on the same value twice.

i.e. f(x₁) != F(x₂) whenever x₁ != x₂

Question 7

Define an inverse function

Answer 7

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

Its 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

f and f^-1 are symmetrical about the line

Answer 8

y = x

Question 9

Theorem 6 in continuity of inverses

Answer 9

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

Question 10

Theorem 7 on differentiability of inverses

Answer 10

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

Question 11

Define the natural logarithmic function

Question 12

The derivative of lnx

Question 13

3 Laws of logarithms

Answer 13

If x,y are positive numbers and r a rational number, then:

Question 14

Steps to logarithmic differentiation:

Answer 14

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

Question 15

Inverses of the natural Exponential function:

Answer 15

• e^{ln<strong>x</strong>}** **= x , x>0
• ln(e^x) = x for all x

Question 16

Properties of e^x

Answer 16

• Increasing continuous function
• Domain e^x = {x E R}
• Range = { y>0 }
• lim x-> infinity^{<strong>– </strong>} e^x = 0,
• lim x-> infinity⁺ e^x = infinity;
  
  • x-axis is horizontal asymptote

Question 17

Derivative of e^x

Answer 17

e^x

Question 18

Integral of e^x

Answer 18

e^x + C