The inverse function (Sect. 7.1)
One-to-one functions.
The inverse function
The graph of the inverse function.
Derivatives of the inverse function.
One-to-one functions
Remark:
Not every function is invertible.
Only one-to-one functions are invertible.
De
Review for Midterm Exam 1.
5 or 6 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to webwork.
Midterm Exam 1 covers:
Volumes using cross-sections (6.1).
Arc-length of curves on the plane (6.3).
Work and uid for
Natural Logarithms (Sect. 7.2)
Denition as an integral.
The derivative and properties.
The graph of the natural logarithm.
Integrals involving logarithms.
Logarithmic dierentiation.
Denition as an integral
Recall:
(a) The derivative of y = x n is y = n x
Integration techniques (Supp. Material 8-IT)
Substitution rule.
Completing the square.
Trigonometric identities.
Polynomial division.
Multiplying by 1.
Substitution rule
Theorem
For every dierentiable functions f , u : R R holds,
f (u(x) u (x) dx =
f (y )
The arc-length of curves in the plane (Sect. 6.3)
The main arc-length formula.
Curves with vertical asymptotes.
The arc-length function.
The main length formula
Remark: The length of a straight
segment can be obtained with
Pythagoras Theorem.
L=
(x)2 + (y
The exponential function (Sect. 7.3)
Review: The exponential function e x .
Computing the number e.
The exponential function ax .
Derivatives and integrals.
Logarithms with base a R.
Review: The exponential function e x
y = exp (x)
y
Denition
The exponent
Volumes as integrals of cross-sections (Sect. 6.1)
The volume of simple regions in space
Volumes integrating cross-sections:
The general case.
Regions of revolution.
Certain regions with holes.
Volumes as integrals of cross-sections (Sect. 6.1)
The volume
Inverse trigonometric functions (Sect. 7.6)
Today: Denitions and properties.
Domains restrictions and inverse trigs.
Evaluating inverse trigs at simple values.
Few identities for inverse trigs.
Next class: Derivatives and integrals.
Derivatives.
Anti-deri
Work on solids and uids (Sect. 6.5)
Moving things around.
Forces made by springs.
Pumping liquids.
Moving Things around: Constant forces
Remarks:
Moving things around requires some work.
Work is an amount of energy needed to move an object.
Remark: If an
The exponential function (Sect. 7.3)
The inverse of the logarithm.
Derivatives and integrals.
Algebraic properties.
The inverse of the logarithm
Remark: The natural logarithm ln : (0, ) R is a one-to-one
function, hence invertible.
y = exp (x)
y
Denition
Hyperbolic functions (Sect. 7.7)
Circular and hyperbolic functions.
Denitions and identities.
Derivatives of hyperbolic functions.
Integrals of hyperbolic functions.
Circular and hyperbolic functions
Remark: Trigonometric functions are also called circula