Differential Calculus

Question 1

Who are the people who devised the differential calculus?

Answer 1

Isaac Newton and G.W Leibniz

Question 2

concerned with the problem of finding the rate of change of a function

Answer 2

Differential Calculus

Question 3

_____ is the value the function gets arbitrarily close to as the function approaches an arbitrary value, a.

Answer 3

limits of function

Question 4

f(x) negative side of zero

Answer 4

-0.00001

Question 5

f(x) positive side of zero

Answer 5

0.00001

Question 6

value of limit as x approaches negative infinity

Answer 6

-99999

Question 7

value of limit as x approaches positive infinity

Answer 7

99999

Question 8

lines that continually approach a given curve but does not meet at any finite distance

Answer 8

asymptote

Question 9

What is the condition in getting the vertical asymptote?

Answer 9

equate denominator to zero to get the x eqn

Question 10

What are the condition in getting the horizontal asymptote

Answer 10

If:
m=n ; y= an/bm
m<n ; no horizontal asymptote
m>n ; y=0

Question 11

What is the condition in getting the inclined asymptote

Answer 11

when m<n , or when there is no horizontal asympote

Question 12

How to get the eqn of the inclined asymptote

Answer 12

using limit approach; use eqn y=mx+c
where:
m= lim as x approaches positive or negative infinity= f(x)/x
c= lim as x approaches positive or negative infinity= f(x)-mx
then substitute m and c to y=mx+c

Question 13

d/dx (u^n)

Answer 13

n . u^n-1

Question 14

d/dx (uv)

Answer 14

vdu+udv

Question 15

d/dx (u/v)

Answer 15

(vdu-udv)/v^2

Question 16

d/dx (e^u)

Answer 16

e^u. du

Question 17

d/dx (ln u)

Answer 17

du/u

Question 18

d/dx (sin X)

Answer 18

cos x

Question 19

d/dx (cos x)

Answer 19

-sin x

Question 20

d/dx (tan x)

Answer 20

sec^2 x

Question 21

d/dx (cot x)

Answer 21

csc^2 x

Question 22

d/dx (sec x)

Answer 22

sec x tan x

Question 23

d/dx (csc x)

Answer 23

-csc x cot x

Question 24

how to solve implicit differentiation

Answer 24

differentiate x and y but add y’ to y, then isolate y’ to left side to get the eqn

Question 25

implicit differentiation caltech: format / formula

Answer 25

B-A / 1X10-5

Question 26

implicit differentiation caltech steps

Answer 26

1. input the eqn in the calculator, do not forget to put [,Y], then let x= 0.1
2. Store value to A
3. input the eqn in the calculator, do not forget to put [,Y], then let x= 0.1+1x10-5
4. Store answer to B
5. Apply format/formula
6. For checking: input the choices in the calcu and let y be the stored value Y

Question 27

Higher derivatives caltech

Answer 27

use d/dx function on calcu, let x= -1 and -1+1x10-5
apply to format/formula

B-A / 1X10-5

Question 28

Motion problems: What is the relationship between s, v , and a

Answer 28

v= ds/dt
a= dv/dt

Question 29

Tangent Line eqn

Answer 29

1. use implicit differentiation to make gen eqn to std eqntp get m
2. use shortcut in creating the eqn (see notebook for full explanation)

Question 30

Normal Line eqn

Answer 30

1. Do the steps in getting the tangent line eqn first
2. Apply the shortcut for normal line eqn (see notebook for full explanation)

Question 31

Slope of the curve

Answer 31

1. use implicit differentiation, then substitute P(x,y)

Question 32

Critical points of the curve

Answer 32

1. Take the first derivative of the eqn
2. get x1 and x2 using the quadratic eqn
3. substitute x1 and x2 to the original eqn to get y1 and y2

Question 33

Inflection point

Answer 33

1. take the second derivative of the eqn
2. get x1 and x2 using the quadratic eqn
3. substitute x1 and x2 to the original eqn to get y1 and y2

Question 34

what is the order of differentiation in getting the slope of the curve?

Answer 34

1st order

Question 35

what is the order of differentiation in getting the inflection point

Answer 35

2nd order

Question 36

what is the order of differentiation in getting the critical point

Answer 36

1st order