Calculus II-Stewart
Dr. Berg
Spring 2010
4.9
Antiderivatives
In many problems we wish to find a function that has a certain rate of change. As an example, given the velocity function of a particle (and a little more information) we can reconstruct the pos
M408L Final Exam Review Day 2
1) Partial Derivatives
A) Find the slope in the x-direction at the point (1, 0, f (1, 0)
on the graph of f when
f ( x, y ) x3 y ln x 2 y
B) Find f xy when
f ( x, y ) e x sin( xy )
2
2) Double Integrals
A) Calculate the value
M408L Final Exam Review Day 1
1)
A car is traveling 60 mi/hr when the brakes are fully applied, skidding the car to a stop at
a constant deceleration of 22 ft / sec2 . What is the length of the skid mark on the
highway? (Hint: 60 mi/hr = 88 ft/sec)
2)
A)
Calculus II-Stewart
Dr. Berg
Spring 2010
8.3
Trigonometric Substitution
We use trigonometric identities to algebraically simplify functions to be
integrated. For example,
x
1 x 2 dx can be easily integrated using the method of u
substitution, but the area
Calculus II-Stewart
Dr. Berg
Spring 2010
8.2
Trigonometric Integrals
We use trigonometric identities to integrate certain combinations of trig functions.
Powers of Sine and Cosine
Example A Evaluate
sin
3
x dx .
Solution: We use the basic Pythagorean ide
Calculus II-Stewart
Dr. Berg
Spring 2010
8.1
Integration by Parts
Recall that, by the product rule,
f ( x ) g( x ) = by parts.
f ( x ) g( x )dx +
d [ f ( x )g( x )] = f ( x )g( x ) + f ( x ) g( x ) . Therefore dx f ( x ) g( x ) dx . This yields the resul
Calculus II-Stewart
Dr. Berg
Spring 2010
7.6
Inverse Trig Functions
Inverses
The inverses of the trig functions are an important addition to our toolbox. Recall that f 1 ( x ) = y if and only if f ( y ) = x , and that to be invertible, a function must be
Calculus II-Stewart
Dr. Berg
Spring 2010
7.4
Logarithmic Functions
Derivatives
Theorem
d dx
d (ln x ) = x and dx (ln g( x ) ) = g( x ) g( x ) .
1
1
Example A
d dx
(ln (tan x ) = tan x sec 2 x .
1
Example B
d dx
(ln x
3
2x =
)
1 (3x 2 2) . x 2x
3
Antider
Calculus II-Stewart
Dr. Berg
Spring 2010
7.2
Calculus with Exponential Functions
The Exponential Function
Bacteria double their population at regular intervals through cell division. We speak of the half-life of radioactive material. Exponential functions
Calculus II-Stewart
Dr. Berg
Spring 2010
6.2
Volume
Volume by Cross Section
A right cylinder has congruent cross sections and side(s) at right angles to the base. A prism, for example, is a right triangular cylinder. If the area of a cross section is A an
M408L
Integral Calculus
Dr. Berg
6.1
Area Between Curves
To approximate the area between two curves, we can use rectangles as we did before. Suppose that f ( x ) g( x ) on the interval [ a, b] . Then, if we divide the interval ba into n intervals of equal
Calculus II-Stewart
Dr. Berg
Spring 2010
5.5
d dx
The Substitution Rule
The substitution rule reverses the chain rule. Recall that, by the chain rule, [ f (g( x )] = f ( g( x ) g( x ) . Reversing this yields d f (g( x )g( x ) dx = dx [ f (g( x )] dx = f (
Calculus II-Stewart
Dr. Berg
Spring 2010
5.4
Indefinite Integrals and the Net Change Theorem
Antiderivatives play such an important role in integration that integral notation is commonly used to represent them.
Definition
f ( x ) dx = F ( x ) means
d dx
F
Calculus II-Stewart
Dr. Berg
Spring 2010
5.3
The Fundamental Theorem of Calculus
A revolution took place in calculus when it was realized by Isaac Barrow (16301677), who was Newtons teacher. that antiderivatives could be used to calculate definite integra
Calculus II-Stewart
Dr. Berg
Spring 2010
5.2 The Definite Integral
We saw in the previous section that taking limits of estimating sums can be used to find area and distance. Now we generalize the ideas involved.
Definition
If f is a function defined on t
Calculus II-Stewart
Dr. Berg
Spring 2010
5.1
shapes.
Areas and Distances
Integral calculus arose from the need to calculate areas and volumes of complex
The Area Problem
Before the Fundamental Theorem of Calculus was discovered, the standard method of cal