Optimization
Write one keyword that is commonly used in optimization problems.
Possible keywords include
1) Minimize
2) Maximize
3) Largest
4) Smallest
5) Optimial
What is the domain of a function? How do you find the domain?
The domain is the set of inputs a function can take.
To find the domain, find any points of restriction (0 in denominator or negatives in square roots are some examples).
What is the deriative of ln(x)?
(ln(x))' = 1/x
Why do we need to restrict the domain of arcsin(x)?
Because sin(x) is not one-to-one!
What were some previous methods to find the limit of a function?
1) Factor
2) Conjugate
3) Multiply by 1/highest power
4) Squeeze theorem
5) lim x-> 0 sin(x)/x = 1
What is a critical point for a function f(x)?
Where f'(c) = 0 or DNE such that c is in the domain.
How do you find the inverse of a function y = f(x)?
To find the inverse, interchange x and y, solve for y, and let y = f^(-1)(x).
What is ln(ab)? ln(a/b)? ln(a^b)?
ln(ab) = ln(a) + ln(b)
ln(a/b) = ln(a) - ln(b)
ln(a^b) = bln(a)
What is the formula for population growth or exponential decay or compound interest?
X(t) = X0e^(kt) where X0 is the initial population/amount and k is the growth/decay/interest rate.
Note: I choose X(t) and X_0 to keep things general! In Anna's notes, for populations you would have P(t) and P0, for exponential decay you would have M(t) and M_0 and for interest, you would have A(t) and A_0.
What is L'Hopital's rule?
Suppose that f(x) and g(x) are differentiable and g'(x) != 0. If lim x->a f(x)/g(x) = 0/0 OR inf/inf, then lim x->a f(x)/g(x) = lim x-> a f'(x)/g'(x).
What is the first derivative test?
Suppose a is a critical point for a function f.
1) If f'(a) changes from + to - at a, then a is a local maximum
2) If f'(a) changes from - to + at a, then a is a local minimum
3) If f'(a) does not change sign, then a is not a local extrema.
Consider the one-to-one function f(x) with domain A and range B. Let g(x) be the inverse of f(x), i.e. g(x) = f^(-1)(x).
What is the domain and range of g(x)?
The domain of g(x) is B and the range is A.
What is a^(x+y)? a^(x-y)? a^(x)^(y)? (ab)^x?
a^(x+y) = (a^x)(a^y)
a^(x-y) = (a^x)/(a^y)
a^(x)^(y) = a^(xy)
(ab)^x = (a^x)(b^x)
How would you go about finding the value for x = arctan(sqrt(3))?
We can write x = arctan(sqrt(3)) as tan(x) = sqrt(3). Then all we need to do is find the x values on the unit circle that satisfy the equation!
1) 0*inf
2) inf - inf
3) 0^0
4) 1^inf
5) inf^0
What is the second derivative test?
Suppose a is a critical point for a function f.
1) If f''(a) > 0, then a is a local minimum
2) If f''(a) < 0, then a is a local maximum
If f(x) is one-to-one and f and f^-1 are both differentiable, what is (f^-1(x))'(b)?
(f^-1(x))'(b) = 1/f'(a) where b = f(a).
What is the setup of logarithmic differentiation? When is it helpful?
The set up looks as follows:
For y = f(x), we find ln(y) = ln(f(x)) and implicitly differentiate. So, (1\y)(dy\dx) = (1\f(x))(f'(x)). The last step would be to multiply y over.
Log diff is helpful when the function of interest is full of products and quotients. Applying a log to a function like this will turn the products into sums of logs and the quotients into differences of logs, which is way easier to take a derivative of!
What is the formula for Netwon's Law of Cooling?
T(t) = T_s + (T_0 - T_s)e^(kt) where T_s is the surrounding temp, k is the heating/cooling rate, and T0 is the initial temp.
What is the methods to get an indeterminate product in the right form to use L'Hopitals?
For products, you can rewrite lim x->a f(x)g(x) as lim x->a (f(x))/(1/g(x)) OR lim x->a (g(x))/(1/f(x)).
What is the area of a circle? The circumference of a circle? The volume of a cylinder? The surface area of a cylinder?
Area of circle: pi*r^2
Circumference of circle: 2*pi*r
Volume of cylinder: h*pi*r^2
Surface Area of cylinder: 2*pi*r^2 + 2*pi*r*h
What does it mean for a function to be one-to-one? What is an important result of a function being one-to-one? How can you prove a function is one-to-one?
We have two definitions
1) A function is one-to-one if it passes the vertical and horizontal line test
2) A function is one-to-one if, for every input, there is a unique output
An important result of a function being one-to-one is that it tells us an inverse exists.
To prove a function is one-to-one, it is sufficient to show that the function is strictly increasing or decreasing. You can also use the definition.The change of base formula is as follows:
log_a(b) = log_c(b) / log_c(a) where c is whatever base we want it to be.
Extra note: the change of base formula is helpful since we can change it to whatever base we want. Usually, we like base e (natural log) since it is the most familiar to us!
How would you go about simplifying sin(arctan(x))?
First, set theta = arctan(x). Then, tan(theta) = x or x/1 and draw a reference triangle. Using the arctan(x)a Pythagorean theorem, we have that theta = sqrt(1 + x^2). So, sin(arctan(x)) = sin(theta) = x/sqrt(1+x^2).
What is the methods to get an indeterminate power in the right form to use L'Hopitals?
In short, use logs!!!
Mathematically, set y = lim x->a f(x)^g(x). We can rewrite this as ln(y) = lim x->a g(x)ln(f(x)) = lim x->a ln(f(x))/(1/g(x)).