Math 121 Final Exam Review Questions fully solved & verified for accuracy(A+graded)

How can we make a quick table of values for sine and cosine?

Square roots over 2: sin(0)=SQRT(0)/2, sin(Pi/6)=SQRT(1)/2, etc.

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec^2(x)

Antiderivative of sin(x)

-cos(x)

Antiderivative of cos(x)

sin(x)

Antiderivative of sec^2(x)

tan(x)

sin(0)

0

sin(Pi/4)

SQRT(2)/2 or 1/SQRT(2)

sin(Pi/2)

1

cos(Pi/6)

SQRT(3)/2

cos(0)

1

cos(Pi/4)

SQRT(2)/2 or 1/SQRT(2)

sin(Pi/3)

SQRT(3)/2

cos(Pi/2)

0

sin(Pi/6)

1/2

cos(Pi/3)

1/2

How do we know if we should use substitution?

We are multiplying by the derivative of something else.

What does the integral from substitution look like?

f(x)dx -> g(u(x))u'(x)dx -> g(u)du

Which derivative do we need to take for substitution?

du

Which things should we rule out before using integration by parts?

Antiderivative rules and substitution.

What is the formula for integration by parts?

[Integral] f(x)g(x)dx = f(x)G(x) - [Integral] f '(x)G(x)dx

How do we choose f(x) and g(x)?

Pick f and g to let f '(x) and G(x) get rid of problematic terms.

What should we make sure actually exists for integration by parts?

G(x) = [Integral] g(x)dx

Which three sums can we use to approximate [Integral] f(x)dx?

Midpoint, Trapezoidal, and Simpson's Rules.

How do we determine [Delta] x?

[Delta] x = (b-a)/n

What points do we plug in to f(x) for the Trapezoidal Rule?

a0 = a, a1 = a0+[Delta]x, a2 = a1+[Delta]x, etc. up to b

What is the formula for the Trapezoidal Rule?

[Delta]x/2 * (f(a0)+2f(a1)+2f(a2)+...+2f(a[n-1])+f(b))

What points do we plug in to f(x) for the Midpoint Rule?

x0 = (a0+a1)/2, x1 = (a1+a2)/2, etc. up to x[n] or x1 = a+[Delta}x/2, x2 = x1+[Delta]x, x3 = x2+[Delta]x, etc. up to x[n]

What is the formula for the Midpoint Rule?

[Delta]x * (f(x1)+f(x1)+f(x2)+...+f(x[n]))

What is the formula for Simpson's Rule in terms of M and T?

S=(2M+T)/3

What change do we have to make to improper integrals?

[Integral] from T to infinity -> lim(b to infinity)[Integral] from T to b

How do we know what a fraction does in the limit?

Exponentials grow fastest, then powers of x, then things like logarithms.

How can we verify the solution to a differential equation?

Take the necessary derivatives, plug them in to the equation, and check that you end up with the correct result.

What is the 'separation' in Separation of Variables?

Factoring the differential equation.

What is the standard form for Separation of Variables?

y ' = f(t) * g(y)

Once we have factored, what's the next step in Separation of Variables?

Put all the 'y's on one side, all the 't's on the other, and integrate both sides.

What are the defining characteristics of a 1st order linear differential equation?

1st order: only one derivative (y ' ), linear: The only power of y is y^1

What is the standard form for a 1st order linear differential equation?

y ' +a(t)y = b(t)

How do we get the integrating factor from the standard form for 1st order linear differential equations?

IF = e^[ A(t) ] , where A(t) is the antiderivative of a(t)

What do we do with the integrating factor for 1st order linear differential equations?

e^[ A(t) ] y = [Integral] e^[ A(t) ] b(t) dt

When is a differential equation autonomous?

When y ' does not depend on t

What two coordinate planes do we use for sketching solutions to autonomous differential equations?

ty (t horizontal, y vertical) and yz (y horizontal, z = y ' vertical)

What kind of solution do we look for right away with autonomous differential equations?

Constant solutions, y ' = 0

How do other solutions to autonomous differential equations relate to the constant solutions?

Solutions to autonomous differential equations are always heading to or starting from constant solutions. Every solution either goes off to infinity or has an asymptote at a constant solution.

Which direction on the yz plane are we moving if y ' is positive?

Right: y is the horizontal axis, so y ' > 0 means that we are moving in the positive direction on the y axis.

Which direction on the yz plane are we moving if y ' is negative?

Left: y is the horizontal axis, so y' < 0 means that we are moving in the negative direction on the y axis.

Which coordinate does Euler's Method give us?

y : Euler's Method is a means for estimating future values for differential equations.

What is the formula for Euler's Method?

y(t + [Delta t]) = y(t) + y ' (t) * [Delta t]

How far do we move horizontally with Euler's Method?

We move right by [Delta t] units each time, with [Delta t] = (b-a)/n

How far up do we move with Euler's Method?

Rise = slope horizontal change, so we move up by y ' (t) [Delta t] at each step.

What is a series?

An infinite sum.

What is a power series?

A series whose terms have powers of x : the nth term is a(n)*x^n

How can we identify a geometric series?

Any series that changes by some fixed factor at each step is geometric, so we look for a constant ratio from one term to the next.

When does a geometric series converge?

When | r | < 1 , where r is the factor that multiplies each successive term.

If a geometric series converges, what is its value?

a/(1 - r) , where a is the first term in the sequence and r is the factor multiplying each successive term.

What is the definition of the third Taylor polynomial of f(x)?

p_3 (x) = f(0) + f ' (0)/1!x^1 + f '' (0)/2!x^2 + f ''' (0)/3!*x^3

How many powers of x can the nth Taylor polynomial of f(x) have?

n Important note: coefficients can be zero, so there might be fewer.

How many derivatives of f(x) does the nth Taylor polynomial of f(x) have?

Exactly n

How can we use the third Taylor polynomial of f(x) to estimate the integral of f(x)?

Take the integral of p_3 (x), instead, since it's a polynomial and we can do it by hand.

How can we use the 2nd Taylor polynomial to estimate the value of f(x)?

Compute p_2(x) instead of f(x), since the former is a polynomial that we can compute by hand.

How many powers of x can a Taylor series have?

Infinitely many.

How many derivatives of f(x) does its Taylor series have?

All of them: f(0), f ' (0), f '' (0), f ''' (0), ...

What is the definition of the Taylor series of f(x)?

The sum from n = 0 to infinity of [the nth derivative of f ](0)/n! * x^n

What changes can we make to a Taylor series to get the Taylor series of another function?

We can add, subtract, multiply, and divide by powers of x, so long as we don't end up with negative powers. We can also plug in powers of x to the function ( f( x^n ) ) and integrate or differentiate the power series.

What is the Taylor series of the integral?

The integral of the Taylor series.

What is the Taylor series of the derivative?

The derivative of the Taylor series.

What is the integral of the Taylor series?

The Taylor series of the integral.

What is the derivative of the Taylor series?

The Taylor series of the derivative.

What is the Taylor series of 1 / (1 - x)?

1 / (1 - x) = 1 + x + x^2 + x^3 + ...

What is the Taylor series of e^x?

e^x = 1 + 1/1! x^1 + 1/2! x^2 + 1/3! x^3 + ...

What is the Taylor series of sin(x)?

sin(x) = x - 1/3! x^3 + 1/5! x^5 - 1/7! x^7 + ...

What is the Taylor series of cos(x)?

cos(x) = 1 - 1/2! x^2 + 1/4! x^4 - 1/6! x^6 + ...

How can we tell apart the Taylor series for sin(x) and cos(x)?

cos(x) is an even function with even powers and sin(x) is an odd function with odd powers. Alternatively, check the leading term: cos(0) = 1, so the Taylor series for cosine starts with 1, while sin(0) = 0, so the Taylor series for sine starts with x.

How many conditions does the Integral Series Test have?

Three.

What are the conditions f(x) must satisfy for the Integral Series Test to apply?

f(x) must be positive, continuous, and decreasing.

What does the Integral Series Test tell us about a series?

If the test applies, then the series converges if the integral converges and the series diverges if the integral diverges.

What does the Integral Series Test not tell us?

The Integral Series Test does not give us a value, even if the series converges.

How do you apply the Integral Series Test, once it applies?

Take the f(n) or f(k) from the series and check if the integral of f(x) converges.

What is the relationship between the probability density function and the cumulative distribution function?

The cumulative density function is the integral of the probability density function.

What are the limits of integration for the cumulative distribution function F(x)?

A and x : the left endpoint of the domain to the current point.

Given the cumulative distribution function F(x), how can we recover the probability density function?

f(x) = F ' (x)

What in the name of the cumulative distribution function clues us in to whether it is f(x) or F(x)?

'Cumulative' tells us that it's adding up from the beginning, ie an integral, so it is F(x).

What in the name of the probability density function clues us in to whether it is f(x) or F(x)?

'Probability density' tells us that it is the density of the probability, ie a derivative, so it is f(x).

At which points x do we always know the value of the cumulative distribution function F(x)?

A and B, the two endpoints of the domain.

What are the values of the cumulative distribution function F(x) at its known points?

F(A) = 0, since there's been nothing to accumulate, and F(B) = 1, since probabilities always sum to 1.

What are the two ways to compute the probability of a continuous random variable?

P(a <= X <= b) = F(b)-F(a), with the cumulative distribution function, and P(a <= X <= b) = the integral from a to b of f(x)dx, with the probability density function.

If we compute P(a <= X <= b), what numbers do we plug into the cumulative distribution function F(x)?

a and b : P(a <= X <= b) = F(b) - F(a)

If we compute P(a <= X <= b) from the probability density function f(x), what numbers do we use as the bounds for the integral?

a and b : P(a <= X <= b) = [Integral from a to b] f(x) dx

What does probability always sum to?

1

What is the expected value of a continuous random variable?

E(X) = [Integral from A to B] x * f(x) dx

What is the variance of a continuous random variable?

Var(X) = [Integral from A to B] x^2 * f(x) dx - [E(X)]^2

What are the bounds on the integral for the expected value?

A and B, the two endpoints of the domain, since the expected value is a number and not a function.

What are the bounds on the integral for the variance?

A and B, the two endpoints of the domain, since the expected value is a number and not a function.

Will the bounds be the same in the integral for probability as they are in the integral for expected value or variance?

No : probability is on an arbitrary range, while expected value and variance cover everything.

Which distributions do we need to be able to recognize and extract information from?

Exponential, Poisson, Normal, and Geometric.

Which two of our known distributions are continuous?

Exponential and normal : the two with probability density functions f(x)

Which two of our known distributions are discrete?

Poisson and Geometric : the two with P(X = n) formulas

Which distribution has f(x) = 1/( sigma sqrt(2 Pi)) e^(- (x-mu)^2/(2 sigma^2))?

Normal.

Which distribution has probability P(X=n) = p^n * (1-p)?

Geometric.