BASIC CALCULUS TERMINOLOGIES Flashcards
● one of the most important ideas or concepts in studying calculus
● the backbone of the calculus, both differential and integral calculus.
LIMITS
Ways to determine if the limit exists
○ by making the table of values of the function
○ by graphing the given function.
○ to find out whether the value of x will approach the value of the constant c, we need to specify the direction
One - Sided Limits
Left - sided limits
→ limit from the left, less than c
lim f(x) = L
x → c-
- Right - sided limits
→ limit from the right, greater than c
lim f(x) = L
x → c+
the limit exists since the values from
left and right are equal.
lim f(x) = lim f(x)
x → c- x → c+
limit does not exist since the values
from left & right are not equal.
lim f(x) ≠ lim f(x)
x → c- x → c+
SOLVE:
lim (1 + 2x)
x→ 3
→ 1 + 2 (3) → 1 + 6 = 7
different left and right limits at every integer
Greatest - integer function
limit of a constant is that constant
Theorem on Limit of Functions
○ function values decrease or increase without bounds as independent variable gets closer and closer to a certain fixed number.
Infinite Limits
if x equates to 0 and → f(x) = 1
x
If denominator equate to 0 & numerator is not 0
let exponent a be even and let exponent b be odd
Even and Odd exponents
If the highest power of x is the same in both the
numerator and denominator, then :
VII. Limits of Rational Expression as x → ∞
If the highest power of x is in the denominator…
lim f(x) = 0
x → ∞
If the highest power of x is in the numerator…
lim f(x) = ∞
x → ∞
○ continuous limits — when you can draw the graph with
1 continuous line without removing the tip of the pencil
from the paper.
Continuous Limits
The limit is CONTINUOUS if :
- f (c) is defined
- lim f(x) exists
x → c - lim f(x) = f(c)
x → c
● a function is said to be continuous if you can sketch its graph on a curve without lifting your pen even once.
● a function is continuous at a particular point if there is no break in its graph at that point.
● Polynomial and Absolute Value Functions are
continuous everywhere.
● Rational Functions are continuous in their respective domains.
Continuity
● a function is continuous on an interval if we can trace the graph from start to finish without lifting the pen
● a function is continuous if its graph has no holes or breaks in it.
Intervals
● the function of the graph which is not connected with each other is known as a discontinuous function.
Discontinuity
TYPES OF DISCONTINUITY : (4)
JUMP
POINT
ESSENTIAL
REMOVABLE
○ this discontinuity occurs when graph “breaks” at a particular place and starts somewhere else.
○ In other words, lim f(x) ≠ lim f(x)
x → a + x → a-
JUMP
○ this discontinuity occurs when the graph has a “hole” in it from a missing point because the function has a value at that point that’s “off the graph”
○ In other words, lim f (x) ≠ f (a)
x → a
POINT
○ this discontinuity occurs when the graph has a vertical asymptote.
ESSENTIAL
○ this discontinuity occurs when there is a hole in the graph of the function.
REMOVABLE
— slope of a tangent line
DERIVATIVE
— straight line that is
tangent to a curve and touches a
function at only one point.
TANGENT LINE
a line l passing through distinct
points (X0 , Y0) and (X1 , Y1) has a
slope : → ml = y1 – y0
x1 – x0
● lines are secant lines
Slope of a Tangent Line
SLOPE FORMULA
RISE OVER RUN
FORMS OF A SLOPE OF A TAGENT LINE (3)
POINT SLOPE
SLOPE INTERCEPT
GENERAL EQUATION OF A LINE
○ y - y0 = m (x - x0)
○ y - y1 = m (x - x1)
point slope form
○ y = mx + b
slope-intercept form :
○ Ax + By + C = 0
general equation of a line
RULES IN FINDING THE DERIVATIVE (6)
CONSTANT
POWER
CONSTANT MULTIPLE
THE SUM
PRODUCT
QUOTIENT
○ if f(x) = c where c is a constant, then f
1(x) = 0.
The derivative of a constant is equal to 0.
CONSTANT RULE
if f(x) = xn where n ∈ N, then f1(x) = nx n - 1
POWER RULE
○ if f(x) = k h(x) where k is a constant, then :
then f1(x) = k h1(x)
CONSTANT MULTIPLE RULE
○ if f(x) = g(x) + h(x) where g and h are differentiable functions, then f
1(x) = g1(x) + h1(x)
SUM RULE
○ the derivative of the product is NOT the product of the derivative.
○ If f and g are differentiable functions, then :
→ Dx [f(x) g(x)] = f(x) g
1(x) + g(x) f
1(x)
○ states that the derivative of the product of two differentiable functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.
PRODUCT RULE
let f(x) and g(x) be two differentiable functions with g(x) ≠ 0.
QUOTIENT RULE
is the square of the original denominator
DENOMINATOR
