Calculus

Question 1

What is a derivative

Answer 1

Derivative of a function = dy/dx = slope at a point

It gives us the instantaneous rate of change of a function.
Eg. If derivative of a function = 5x, then the instantaneous rate of change of that function when x=3, would be 15.

Question 2

Why is the derivative of a constant value 0

Answer 2

For a constant function, say c, The value of y remains same as long as x stays constant say “c”
Which means f’(c) = 0

Question 3

Derivative of a constant is always

Answer 3

0 ,because in the graph: when we check the slope at any point is 0.

Question 4

In a graph in which y=x³ the slope at any point in the graph (derivative) :

Answer 4

3x³⁻¹ = 3x²

This rule is called the power rule.

Question 5

When you take f(x) = C (a constant) the slope of it is

Answer 5

So it means that whatever value you give for x , the value of the y (fx) coordinate always has the same value 7. As there is no change in the y coordinate , the slope is 0.

Question 6

The shape of the graph when y = f(x) = x²

Answer 6

Parabola

Question 7

To find the indefinite integral of a f(x) means

Answer 7

Finding the anti derivative of f(x)

That would be another function.

Question 8

Reverse power rule

Answer 8

∫x³ dx = (x ³⁺¹)/3+1 + C

Question 9

An angle in xy plane, is said to be in its standard form, when

Answer 9

The vertex is at the origin and the initial ray lies also get the positive x axis.

Question 10

Sin of 37

Answer 10

3/5

Question 11

Sin of 53

Answer 11

4/5

Question 12

Cos of 37

Answer 12

4/5

Question 13

Tangent

Answer 13

Straight line touching a curve at a particular point.

Question 14

If c is a constant d/dx of c =

Answer 14

0

Question 15

Derivative of x³ is

Answer 15

3x³⁻¹ = 3x²

This rule is called the power rule.

Question 16

d/dx (Cx) =

Answer 16

C d/dx (x)

This rule is called constant multiple rule

Question 17

The derivative of the negative of a differentiable function is the negative of the function’s derivative

Answer 17

d/dx (-u) = -1 d/dx(u)

Question 18

Derivative of the sum of 2 differentiable functions is the sum of their derivatives.
This rule is called

Answer 18

The sum rule:

d/dx(u-v) = d/dx(u) - d/dx(v)

Question 19

In prime notation :

Answer 19

‘ (derivative)
″ (double derivative)
‴ (third derivative)
⁗ (fourth derivative)

Question 20

Product rule

Answer 20

d/dx (uv) = u. d/dx (v) + v. d/dx (u)

Question 21

Derivative of sin x

Answer 21

Cos x

Question 22

Derivative of cos x is

Answer 22

• sin x.

Question 23

Derivative of e raised to x

Answer 23

e raised to x

Question 24

Derívate of the natural log of x =

Answer 24

1/x

Question 25

Chain rule :

Answer 25

``` y = f(g(x)) Then f(x)/f(y) = f’(g(x)) . g’(x) ```

Question 26

Chain rule is also known as

Answer 26

Outside inside rule

Question 27

d/dx of (dy/dx) is written

Answer 27

d²y/dx²

Question 28

At maxima

Answer 28

dx/dy = 0 | d²x/dy² < 0

Question 29

In minima d²x/dy² is

Answer 29

Positive

Question 30

d/dx (Cos kx)/k

Answer 30

Sin kx

Question 31

2 important trigonometric formulas

Answer 31

``` sin²x = (1 - cos2x)/2 cos²x = (1 + cos2x)/2 ```

Question 32

Sin a x cos b =

Answer 32

1/2 [sin(a+b) + sin(a-b)]

Question 33

Cos A x sin B

Answer 33

1/2 x [sin(a+b) - sin(a-b)]

Question 34

In a graph the area is equal to

Answer 34

Area of many small strips. Area of each small strip = f(x)dx Sum of total area of all strips = ∫f(x)dx (Actually it is a definite integral)

Question 35

d(u/v) /dx =

Answer 35

Quotient rule : | (v.du/dx - u.dv/dx) / v²

Question 36

Cos²x - sin²x =

Answer 36

Cos (2x)

Question 37

Derivative of tanx

Answer 37

Sec²x

Question 38

Derivate of sec x

Answer 38

Sec x. tan x

Question 39

Derivative of cosec x

Answer 39

Cosec x. Cot x

Question 40

Chain rule also called

Answer 40

Outside inside rule.

Question 41

Inverse operation of differentiation

Answer 41

Integration

Question 42

Anti derivative of x²

Answer 42

(x²⁺¹) / (2+1) + C

Question 43

Anti derivative of 1

Answer 43

x + C

Question 44

3 conditions for a physical qty to be a vector

Answer 44

Magnitude Direction Should obey laws of vector algebra.

Question 45

Definite integral is used for

Answer 45

Calculation of area of a curve.

Question 46

Derivative gives us

Answer 46

The instantaneous rate of change of a function. Example : if derivative of a function f(x) is 3x It means the instantaneous rate of change of that function when x=7, would be 3x7 = 21.

Question 47

Tan 53

Answer 47

4/3

Question 48

Tan 37

Answer 48

3/4