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Calculus Flashcards

To understand differentiation and integration more effectively

1
Q

Critical points are points in the curve where the slope is

A

zero

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2
Q

All critical points are not Local maxima|Local mininma, but all local maxima and local minima has a critical point why

A

Criticial points can also indicate inflection points

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3
Q

What does y = x2 gives as curve

A

A Parabolic Curve

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4
Q

What does y = x2 gives as curve

A

Similar to a Asymptode

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5
Q

What is Differentiation tells you

A

Slope of a tangent line

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6
Q

What does integration used for

A

To find AUC and average rate of change of moving objects

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7
Q

What are the ways to find slopes

A

If lines - y = mx+c
if curves - Differentiation (Slope of tangent line)
Slope of Secant line

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8
Q

What is a Secant Line

A

Secant Line connects two points in a curve

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9
Q

Slope of a Secant Line

A

y2 - y1/ x2 - x1

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10
Q

how do we find instantaneous rate of change in terms of calculus

A

Differentiation

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11
Q

how do we find instantaneous rate of change in terms of calculus

A

Integration

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12
Q

How do we find acceleration in terms of Calculus

A

second derivative of the function

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13
Q

What is actually a function

A

A relation between Y value and X value

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14
Q

What is the difference between f(x) and F(x)

A

f(x) - differentiation,
F(x) - integration

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15
Q

What is the difference between f’(x) and f^-1(x)

A

f’(x) = differentiation
f^-1(x) = Inverse function

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16
Q

What is the use of this f^-1(x)

A

gives you the value of x wrt to y

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17
Q

What is sine wave

A

Starts at 0, peaks at 90, bottoms at 270 and again peaks at 360

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18
Q

What is Cosine Wave

A

Starts at 90 peaks at 0, bottoms at 180 and again peaks at 270

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19
Q

f(sinx)

A

cosx

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20
Q

f(cosx)

A

-sinx

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21
Q

f(tanx)

A

sec2x

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22
Q

f(cotx)

A

-cosex^2x

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23
Q

f(secx)

A

secx tanx

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24
Q

f(cosecx)

A

-cosecx tanx

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25
Q

What is Power rule in Differentiation

A

n^x = nx^n-1

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26
Q

What is chain rule

A

dy/dx(f(g’(x)) = f’(g(x) * g’(x)
outside first inside second

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27
Q

differentiate f(x) g(x)

A

f’(x) g(x) +f(x) g’(x)

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28
Q

differentiate f(x)/g(x)

A

f’(x) g(x) +f(x) g’(x)/(g(x))^2

29
Q

differentiate c

A

0

30
Q

differentiate x

A

1

31
Q

What is LCD

A

Multiply the numerator and denominator by a common number so that the denominators match for easy calculations

32
Q

What happens when we have bounds of Secant line infinitely close to each other

A

This resembles Tangent Line

33
Q

What is the limit x->2 when y = x2 means

A

it means the the value of x is approaching Ly close to 2 it can be 2.1, 2.001

34
Q

What does Squeeze theorem

A

if b(x) lies between a(x) and c(x) and
differentiation of a(x) and b(x) = d.
then c(x) is also zero

35
Q

What are continuous curve and non continuous cutves

A

Non continuous curve has a break/missing(cannot be drawn in a single hand)
Continuous can

36
Q

What are continuous curve and non continuous cutves

A

Non continuous curve has a break/missing(cannot be drawn in a single hand)
Continuous can

37
Q

Give an example for non continuous curves

A

Jump discontinuity, asymptodes

38
Q

What is an odd function and and even function

A

Odd function is symmentric across either x axis or y axis even function is symmetric across origin

39
Q

What is an odd function and and even function

A

Odd function is symmentric across either x axis or y axis even function is symmetric across origin

40
Q

What is domain and bounds

A

Domains are the highest and lowest values of x-axis and bounds are the highest and lowest values of y axis

41
Q

How do you find concavity of a curve

A

Second derivative of a function gives the distance from the critical point

42
Q

What does it mean if the second derivative of a function is positive

A

It means that the curve is upwards

43
Q

What does it mean if the second derivative of a function is negative

A

The curve is concave down

44
Q

How do you find critical value by Extreme value theorem

A

According to mean value theorem
step 1 take the first derivative of the curve and equate it to zero to find x values
step 2
substitute X value and identify y value from the curve step 3
apply the lower and higher bounds to the equation these are the critical points of the curve

45
Q

When is first derivative preferred over second derivative while calculating critical points

A

If the formula involves fractions taking second derivative leads to complications hence first derivative is used

46
Q

What is jerk

A

It is the third derivative of velocity or second derivative of acceleration

47
Q

What is Local Maxima and Global Maxima

A

local maxima is a peak in relavance to the nearer points, global maxima is the maxima of all the peaks in the curve

48
Q

How can We interpret limits which respective to their values

A

Limits from left = Limit from Right => Then its a asymptode

49
Q

Feauture of Vertical Asymptode

A

y reaches some number, x reaches towards infinity

50
Q

Feauture of Horizontal Asymptode

A

x reaches some number, y reaches towards infinity

51
Q

Roles Theorem

A

If f is continous function
f is differentiable on (a,b)
f(a) = f(b)

52
Q

Mean Value Theorem

A

If f is continous function
f is differentiable on (a,b)
Slope is calculated by y2-y1/x2-x1

53
Q

What is Anti Derivative Called as

A

Integration

54
Q

What is the Use of Integration

A

To Area Under the Curve by summing up infinitely small number of distances

55
Q

i=1 -> n ∑ C =

A

C

56
Q

i=1 -> n ∑ i =

A

n(n+1)/2

57
Q

i=1 -> n ∑ i^2 =

A

n(n+1)(2n+1)/6

58
Q

i=1 -> n ∑ i^3 =

A

n^2(n+1)^2/2

59
Q

What is Riemann Integral

A

Definite Integral

60
Q

What is Definite Integral

A

Definite Integral Has the bounds

61
Q

What is INdefinite Integral

A

Indefinite Integral Has no bounds
∫f(x)dx = F(x) + C

62
Q

How to define Sine, Cos, Tan by trigonometry

A

Sin = opposite/Hypotenuse,
Cosine = Adjacent/Hypotenuse
Tan = Opposite/Hypotenuse

63
Q

What is Newtons Law in calculus used For

A

Newton Law is used to point a point in x axis where y value is zero

64
Q

What is Newtons law of Calculus

A

x(n+1) = xn - ( f(xn)/f’(xn) )

65
Q

Steps in optimization procedure

A

Understand the question
Identify the optimisation equation and the constraints eqn
Now solve the constraints equation to get simpler equation
Substitute this into the optmisation equation and optimize it (ie derivative =0)
Solve it and substitute to constrain eqn and find both values

66
Q

What is optimization in calculus

A

Optimization helps us to maximize or minimise a function
(Ie, finding local maxima or minima)

67
Q

What is optimization in calculus

A

Optimization helps us to maximize or minimise a function
(Ie, finding local maxima or minima)

68
Q

Integration of rate gives

A

Accumulation

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