These are my complete notes for Differential Calculus, covering such topics as Limits, Squeeze Theorem, Velocity, Acceleration, and Position, the Power Rule, Chain Rule, Quotient Rule, Particle Motion, Continuity, Optimization, Related Rates, and more.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Differential Calculus (Complete)
?. Limits.
# Average Rates/Speed: Average Speed is found by dividing the distance covered by the elapsed time.
# Limit: The value a function approaches as it becomes closer to a specified point - thus, not necessarily actually being at that point, but rather as close to it as possible with being on it. If a function is continuous near that point, the limit equals the function's value; otherwise, it will describe the behavior of the function even if undefined at that point.
# Properties of Limits:
Similar to Logarithms, Limits have their internal rules that can applied at any time. The "commandments" version of this list can be seen in Rule 110.
1. The limit of a constant is equal to a constant. $$\lim_{x \to a} c = c$$
2. The limit of x as x approaches a, equals a. $$\lim_{x \to a} x = a$$
3. The limit of a sum is the sum of the limits. $$\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$
4. The limit of a difference is the difference of the limits. $$\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$$
5. The limit of a constant times a function is the constant times to limit of the function. $$\lim_{x \to a} [c × f(x)] = c \lim_{x \to a} f(x)$$
6. The limit of a product is the product of the limits. $$\lim_{x \to a} (f(x) × g(x)) = \lim_{x \to a} f(x) × \lim_{x \to a} g(x)$$
7. The limit of a quotient is the quotient of the limits, provided the denominator does not equal zero. $$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$$
8. The limit of a sum is the sum of the limits. $$\lim_{x \to a} (f(x))^n = (\lim_{x \to a} f(x))^n$$
# Rule 106. The average rate of change IS the slope. The average speed is the distance covered divided by the elapsed time, like miles per hour.
# Rule 107.Say that 16t² is the free fall motion of an object with no air resistance. It will follow the same (∆y / ∆t) formula as the slope formula.
∆ (Delta) means as "the change in", so (change in distance / change in time). From 0 to 2 seconds, the rock will travel (6(2)² - (6(0)^2)) / (2 - 0), or 34 ft/sec average speed.
# Rule 108. To find the instaneous speed/rate of change, the limit formula is as follows: $$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ You know a and the equation at first, plug in (a + h) to the equation and complete formula. Then, plug in 0 for h to find instantaneous velocity/speed.
# Rule 109. Speed is the time rate at which an object is moving along a path, while velocity is the rate and direction of an object's movement.
# Rule 110. There are nine holy truths of limits that must always be remembered and and held immutable. They are similar to the mystical factoring rules, and some are versions of the Limit Properties referenced previously:
- The equivalance of constants. (L.P. #1)
- The equivalance of a to x approaching a. (L.P. #2)
- The Sum Rule. (L.P. #3)
- The Difference Rule. (L.P. #4)
- The Constant Multiplication Rule. (L.P. #5)
- The Limit Product Rule. (L.P. #6)
- The Limit Quotient Rule. (L.P. #7)
- The Limit Power Rule. (L.P. #8)
- The Radical Rule. (a synthesis of L.P. #8 & #7 or #6)
# Rule 111. The one-sided limit is allowed to be infinity, while two-sided limits cannot be infinity and must not be a number.
# Rule 112. The Squeeze Theorem:
When the limit can't be found directly, find it indirectly using the squeeze theorem. The way this theorem works, is to 'squeeze' function f (which has the uncertain limit) between two functions, h and g, and find their limits. Definitionally, this is written as follows:
If g(x) ≤ f(x) ≤ h(x), where x ≠ c in some interval around c, then: $$\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$$
The classic example of a function that must use the Squeeze Theorem is one that oscillates, such as x² × sin(1/x).
The key idea of the Squeeze Theorem is that if h(x), which is above f(x), has its limit at c with the same y-value as g(x), then f(x) would also have to have that limit because it is between g(x) and h(x).
# Infinity/∞: The term 'infinity' does not refer - it describes the behavior of a function. Infinity means an increasing distance forever.
# Rule 113. The squeeze theorem also works for infinite limits.
# Rule 114. To find the horizontal asymptote, you can take the biggest thing on top, and divide it by the biggest thing on the bottom, or you could factor out the biggest base (with exponent), even if it makes every divisor weird. Then, factor to find the vertical asymptote, just plug in zero for the bottom, or cheat by using a graph.
# Rule 115. End Behavior Models
You can determine the end behavior of a complicated polynomial by looking at a similar one that acts exactly the same for extremely large values of x. For all polynomials, g(x) = an × xⁿ is the end behavior.
The function g must be:
a) A right end behavior model for f if $$\lim_{x \to ∞} \frac{f(x)}{g(x)} = 1$$
b) a left end behavior model for f if $$\lim_{x \to -∞} \frac{f(x)}{g(x)} = 1$$
# Rule 116. If the EBM is a constant, then it is a horizontal asymptote.
# Rule 117. A function's right & left end behavior models are not always the same function.
# Rule 118. Tradition dictates that for an equation with only two polynomials, the left one is the REBM and the right is the LEBM. Yes, opposite of what one may immediately think.
# Reciprocal Substitution:
There is a very specific and unique characteristic of Limits that must always be kept in mind when performing problems. Reciprocal substitution can be performed by taking the variable of the limit (90% of the time being x), flip it to its reciprocal (e.g., 1/x for x), and switch the term of the limit as shown below. This reciprocal is performed even for x's inside of trig. functions - it is done for all instances.
$$\lim_{x \to ∞} \frac{1}{x} = \lim_{x \to 0^+} x$$ $$\lim_{x \to -∞} \frac{1}{x} = \lim_{x \to 0^-} x$$
Infinity goes to Zero-Right, as Negative Infinity goes to Zero-Left. The nature of the reciprocal is not specific a specific side - either infinity or the zero could have 1/x or x.
# Rule 119. Rampant idiocracy has softened testing policy enough that you may be able to, in essence, cheat and look at the graph to determine values like the end behavior mode and asymptotes.
# Rule 120. If the assumed number for the LEDM of some two-polynomialed equation doesn't work, then try the answer for the REBM. Same for the issue vice-versa.
# Rule 121. For piecewise functions, even if x is ≥ or ≤ or > or whatever to 0, a (Lim x→0⁻) will make the function go to the f(x) for less than zero, and (Lim x→0⁺) makes it go to x > 0. Think of it like the plus and negative signs -0.1 or add 0.1 to the value.
?. Continuity.
# Rule 122. Any function y = f(x) whose graph y = f(x) can be sketched in one continuous motion without lifting the pencil is an example of a continuous function.
# Rule 123. There are two RULES of CONTINUITY, which if both met, establish UNIVERSAL CONTINUITY for a function.
- A function y = f(x) is continuous at an inerior point c of its domain if:
$$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$$ $$\lim_{x \to c} f(x) = f(c)$$ - A function y = f(x) is continuous at a left endpoint a if:
$$\lim_{x \to a^+} f(x) = f(a)$$ and is continous at a right endpoint b of its domain if: $$\lim_{x \to b^-} f(x) = f(b)$$ If a function f is not continuous at a point x, we say that f is discontinuous at c, and that c is a "point of discontinuity" of f.
# Rule 124. Continuous Function are continuous everywhere on their domains. Properties of Continuous Functions f & g are continuous at x = c.
All polynomial and Radical functions are continuous everywhere with no Points of Discontinuities. On the other hand, all rational and trig. functions are only continuous along their domain; they have Points of Discontinuities wherever the denominator equals 0.
# Operation Terminology Refresher:
- Sums: f + g
- Differences: f - g
- Products: f × g
- Constant Multiples: k × f, where k = any real #.
- Quotients: f/g, g ≠ 0.
# Rule 125. If f is continuous at c, and g is continuous at f(c), then the composite g ∘ f is continuous at c.
# Rule 126. A function is continuous even if it has Points of Discontinuity, as long as these points are not within the domain. For example, 1/x is considered to be continuous along its domain, despite the absurdity of its graph.
# Rule 127. A function passes the Intermediate value theorem if it never takes on two values without taking on every value in between. A continuous function f on a closed interval [a, b] will take on every y-value between f(a) and f(b), given no discontinuities.
# Rule 128. Jump Discontinuities: Where the left & right side limits are not equal.
Infinite Discontinuities: Where the discontinuity is an asymptote.
Removable Discontinuities: A detached dot, floating on the plane, where the limit continues to be a real number.
Oscillating Discontinuities: A discontinuity that oscillates back and forth, son. Examples include sin(1/x). These functions are discontinuities because no matter how far you zoom in, it will still be oscilliating, thus lacking local linearity.
# Rule 129. For Discontinuities: Jump beats hole. Infinite beats Jump. Oscillating is in a league of its own, as it can only be sin(1/x) or something of the sort.
?. Derivative.
# Rule 131. A tangent line is a line that intersects a curve at exactly one point. The slope of the tangent line is the instantaneous rate of change at a point.
# Difference Quotient: $$\frac{f(x + h) - f(x)}{h}$$ The "Difference Quotient", given above, represents the secant line of a curve from the point (a, f(a)) to (a+h, f(a+h)), which is also the average rate of change from x=a to x=a+h:
# Normal Line: The line normal to a curve is the line perpendicular to the tangent at that point.
# Rule 132. When the limit exists, it's called the derivative of f at a, or any point. The derivative is f'(x) = $$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ provided the limit exists. "Differentiate" means to determine the Derivative.
# Verbal Descriptions of Calculus Notation:
# Rule 133. Graphing the derivative without knowledge of the formulas for dy/dx nor the parent function, is wack:
First, get the slope between two points on the original f(x) and find the slope. As x increases, a decreasing f(x) value indicates a negative slope, while an increasing f(x) value indicates a positive slope. Find an x-value in the middle of the two points and use it as the x-value. Use the slope as the y-value and plug into the new graph, and repeat. Fo a graph with "pointy points", known as corners, they are undefined in the derivative graph (the f'(x)), as they do not have local linearity (see [[[).
# Rule 134. When a fraction has a square root, standard procedure is to multiply it by a conjugate, especially if the square root is on the denominator.
# Rule 135. For the alternate definition of the derivative, you always want to do all the algebra first before substituting for a at the end. Then, you can do a.
# Rule 136. Function f(x) is differentiable on a closed interval [a, b] if it has a derivative at every interior point of the interval, and if the limits below exist at the endpoints.
$$\lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$$ is the right-hand derivative, and
$$\lim_{h \to 0^-} \frac{f(b + h) - f(b)}{h}$$ is the left-hand derivative.
# Rule 137. The left hand derivative can be found by using the regular f(x), while the right hand derivative is found by using the equation of the f'(x) derivative found from the left hand derivative. If the two derivatives are equal to each other, then the derivative in totality is real.
# Differentiability: The ability to find slope at a point, which is necessary to take the derivative.
# Cases when f'(a) doesn't exist (or, when a point is non-differentiable):
- A Corner: f(x) = |x|
A corner occurs when, at the point a, the function on the graph will make a sharp turn, and it's slope will change instantaneously.
- A Cusp: f(x) = x2/3
A cusp occurs when the line gradually curves into a sharp corner. In effect, the cusp is a special case of the corner: the cusp only has one possible tangent, while a corner has two distinct ones.
- A Vertical Tangent: f(x) = ∛x
When the line is vertical, there is no slope, and thus it is non-differentiable at that point.
- A Discontinuity:
For example, a non-continuous piecewise function. The derivative (nor the limit, if it is a jump discontinuity) exists at a point that is discontinous.
# Local Linearity: If you focus on a point on a curve, if you continuously zoom in on the point, eventually the graph of the curve at the point will resemble a line. Differentiability implies local linearity.
# Rule 138.
