BASIC CALCULUS TERMINOLOGIES

Question 1

● one of the most important ideas or concepts in studying calculus
● the backbone of the calculus, both differential and integral calculus.

Answer 1

LIMITS

Question 2

Ways to determine if the limit exists

Answer 2

○ by making the table of values of the function
○ by graphing the given function.

Question 3

○ to find out whether the value of x will approach the value of the constant c, we need to specify the direction

Answer 3

One - Sided Limits

Question 4

Left - sided limits
→ limit from the left, less than c

Answer 4

lim f(x) = L
x → c-

Question 5

2. Right - sided limits
   → limit from the right, greater than c

Answer 5

lim f(x) = L
x → c+

Question 6

the limit exists since the values from
left and right are equal.

Answer 6

lim f(x) = lim f(x)
x → c- x → c+

Question 7

limit does not exist since the values
from left & right are not equal.

Answer 7

lim f(x) ≠ lim f(x)
x → c- x → c+

Question 8

SOLVE:
lim (1 + 2x)
x→ 3

Answer 8

→ 1 + 2 (3) → 1 + 6 = 7

Question 9

different left and right limits at every integer

Answer 9

Greatest - integer function

Question 10

limit of a constant is that constant

Answer 10

Theorem on Limit of Functions

Question 11

○ function values decrease or increase without bounds as independent variable gets closer and closer to a certain fixed number.

Answer 11

Infinite Limits

Question 12

if x equates to 0 and → f(x) = 1
x

Answer 12

If denominator equate to 0 & numerator is not 0

Question 13

let exponent a be even and let exponent b be odd

Answer 13

Even and Odd exponents

Question 14

If the highest power of x is the same in both the
numerator and denominator, then :

Answer 14

VII. Limits of Rational Expression as x → ∞

Question 15

If the highest power of x is in the denominator…

Answer 15

lim f(x) = 0
x → ∞

Question 16

If the highest power of x is in the numerator…

Answer 16

lim f(x) = ∞
x → ∞

Question 17

○ continuous limits — when you can draw the graph with
1 continuous line without removing the tip of the pencil
from the paper.

Answer 17

Continuous Limits

Question 18

The limit is CONTINUOUS if :

Answer 18

1. f (c) is defined
2. lim f(x) exists
   x → c
3. lim f(x) = f(c)
   x → c

Question 19

● 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.

Answer 19

Continuity

Question 20

● 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.

Answer 20

Intervals

Question 21

● the function of the graph which is not connected with each other is known as a discontinuous function.

Answer 21

Discontinuity

Question 22

TYPES OF DISCONTINUITY : (4)

Answer 22

JUMP
POINT
ESSENTIAL
REMOVABLE

Question 23

○ 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-

Answer 23

JUMP

Question 24

○ 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

Answer 24

POINT

Question 25

○ this discontinuity occurs when the graph has a vertical asymptote.

Answer 25

ESSENTIAL

Question 26

○ this discontinuity occurs when there is a hole in the graph of the function.

Answer 26

REMOVABLE

Question 27

— slope of a tangent line

Answer 27

DERIVATIVE

Question 28

— straight line that is tangent to a curve and touches a function at only one point.

Answer 28

TANGENT LINE

Question 29

a line l passing through distinct points (X0 , Y0) and (X1 , Y1) has a slope : → ml = y1 – y0 x1 – x0 ● lines are secant lines

Answer 29

Slope of a Tangent Line

Question 30

SLOPE FORMULA

Answer 30

RISE OVER RUN

Question 31

FORMS OF A SLOPE OF A TAGENT LINE (3)

Answer 31

POINT SLOPE SLOPE INTERCEPT GENERAL EQUATION OF A LINE

Question 32

○ y - y0 = m (x - x0) ○ y - y1 = m (x - x1)

Answer 32

point slope form

Question 33

○ y = mx + b

Answer 33

slope-intercept form :

Question 34

○ Ax + By + C = 0

Answer 34

general equation of a line

Question 35

RULES IN FINDING THE DERIVATIVE (6)

Answer 35

CONSTANT POWER CONSTANT MULTIPLE THE SUM PRODUCT QUOTIENT

Question 36

○ if f(x) = c where c is a constant, then f 1(x) = 0. The derivative of a constant is equal to 0.

Answer 36

CONSTANT RULE

Question 37

if f(x) = xn where n ∈ N, then f1(x) = nx n - 1

Answer 37

POWER RULE

Question 38

○ if f(x) = k h(x) where k is a constant, then : then f1(x) = k h1(x)

Answer 38

CONSTANT MULTIPLE RULE

Question 39

○ if f(x) = g(x) + h(x) where g and h are differentiable functions, then f 1(x) = g1(x) + h1(x)

Answer 39

SUM RULE

Question 40

○ 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.

Answer 40

PRODUCT RULE

Question 41

let f(x) and g(x) be two differentiable functions with g(x) ≠ 0.

Answer 41

QUOTIENT RULE

Question 42

is the square of the original denominator

Answer 42

DENOMINATOR