Important Formulas
If f(x) and g(x) are differentiable functions, what formula gives the derivative of their product?
The Product Rule:
f'(x)g(x) + g'(x)f(x)
If the first derivative, f'(x), is positive on an interval, what can you conclude about the function f(x) on that same interval?
The function f(x) is increasing.
If the limit of f(x) as x approaches a from the left is not equal to the limit of f(x) as x approaches a from the right, what can you conclude about the two-sided limit around a?
It does not exist.
What is the integration rule used to find the indefinite integral x^n for any n other than -1?
The power rule for integration, or reverse power rule, or honestly just power rule
What is the slope of the function f(x) = 4x^3 - 2x + 1 at the point where x=1?
f'(x) = 12x - 2, so f'(1) = 10
The definition of the derivative at point a
Limit as a goes to zero of (f(a + h) - f(h)) divided by h
If c is a point in the domain of c where the derivative f'(c) equals zero or f'(c) is undefined, c is known as what kind of point?
A critical point
State the squeeze theorem.
If f(x) >= g(x) >= h(x) for some interval containing a, and f(x) and h(x) have the same limit L at a, then the limit of g(x) at a must also be equal to L.
When finding an indefinite integral, what must you add to the end result to account for the derivative of a potential constant?
We add C to the end result
Find the limit as x approaches 3 of (x^2 - 9)/(x-3)
6
When evaluating the integral of f(x) from a to b, what theorem allows us to equate the definite integral to F(a) - F(b)
If the second derivative, f''(x), is negative on an interval, what can you conclude about the function f(x) on that interval?
The function is concave down.
What are the three conditions that must be met at f(a) for f(x) to be continuous at point a?
1. f(a) is a defined function
2. limit as x approaches a of f(x) exists
3. limit as x approaches a is equal to f(a)
What technique do we employ if we think that the derivative inside of the integral needed that chain rule?
u-substitution
If the velocity of an object is given by v(t) = 2t + 4 (m/s), what is the total distance traveled during the time interval from 0 to 2 seconds?
12 meters
What is the derivative ,with respect to x, of an integral from a constant a to x, where the function inside is f(t)
f(x)
First Fundamental Theorem of Calculus
State the condition for a local maximum at a critical point using the Second Derivative Test.
The sign of f''(c) is negative.
L'Hôpital's Rule can only be applied directly to two specific indeterminate forms. Name both of these forms
0/0 and infinity over infinity.
When we do u-substitution, how does dx come into play?
We need to substitute all x terms in the integral to u, including dx. So we get g(u) = h(x) and take the derivative to find the relationship between du and dx.
Find the x-coordinate where the local maximum occurs for the function f(x) = x^3 - 3x^2 + 1.
x = 0
For a continuous random variable, the probability density function, f(x) is the derivative of the cumulative density function, F(x), which allows you to find the probability f(X < x). State the CDF in terms of PDF f(t)
In statistics and data science regression is a foundational method where we find the best-fit line for a dataset. We often do this by defining and error function equal to the square of the error between our model and the dataset. How can we apply calculus to the error function to find optimal weights?
Take the first derivative and set to zero to find the minimum of the error function.
If f(x) has a corner or a vertical tangent at x=a, and you want the limit of f(x)/g(x) approaches a, what step in the L'Hôpital's Rule fails, and why would this invalidate the rule?
The derivative of f(x) does not exist, and the derivatives of f(x) and g(x) must exist for the rule to work.
Which is greater, the absolute value of a definite integral of f(x), or the definite integral of the absolute value of f(x) and why?
The definite integral of the absolute value of f(x) is greater or equal to the absolute value of the definite integral of x. We add all area, and don't subtract anything.
A 6-foot tall person walks away from a 15-foot high street light at a speed of 2 ft/s. How fast is the tip of the person's shadow moving away from the light pole when the person is 40 feet from the pole?
10/3 ft/s