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JEE Maths > Calculus Basics > Flashcards

Calculus Basics Flashcards

1
Q

What is the derivative of a constant?

A

0

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2
Q

What is the power rule for differentiation?

A

If f(x) = x^n, then f’(x) = n*x^(n-1)

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3
Q

True or False: The derivative of sin(x) is cos(x).

A

True

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4
Q

What is the integral of x^n with respect to x?

A

(x^(n+1))/(n+1) + C, where n ≠ -1

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5
Q

Fill in the blank: The limit of a function as x approaches a point must exist for the function to be ________ at that point.

A

continuous

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6
Q

What is the derivative of e^x?

A

e^x

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7
Q

What is the integral of cos(x) with respect to x?

A

sin(x) + C

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8
Q

What does it mean for a function to be continuous?

A

A function is continuous if it is defined at a point, the limit exists, and the limit equals the function’s value at that point.

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9
Q

True or False: The sum rule of differentiation states that the derivative of a sum is the sum of the derivatives.

A

True

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10
Q

What is the chain rule in differentiation?

A

If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x)

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11
Q

What is the integral of 1/x with respect to x?

A

ln|x| + C

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12
Q

What is the derivative of tan(x)?

A

sec^2(x)

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13
Q

Fill in the blank: The Fundamental Theorem of Calculus connects ________ and ________.

A

differentiation, integration

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14
Q

What is the product rule in differentiation?

A

If u and v are functions, then (uv)’ = u’v + uv’.

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15
Q

True or False: The derivative of ln(x) is 1/x.

A

True

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16
Q

What is the integral of sin(x) with respect to x?

A

-cos(x) + C

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17
Q

What is the quotient rule in differentiation?

A

If u and v are functions, then (u/v)’ = (u’v - uv’)/v^2.

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18
Q

What does the mean value theorem state?

A

If a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f’(c) = (f(b) - f(a))/(b - a).

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19
Q

What is the second derivative test used for?

A

To determine the concavity of a function and to identify local maxima and minima.

20
Q

What is the integral of sec(x)tan(x) with respect to x?

A

sec(x) + C

21
Q

True or False: The limit of a function can be infinite.

A

True

22
Q

What is the derivative of x^3?

A

3x^2

23
Q

Fill in the blank: The derivative represents the ________ of a function at a point.

A

slope

24
Q

What is the integral of e^(ax) with respect to x?

A

(1/a)e^(ax) + C

25
Q

What is the derivative of a sum of functions?

A

The derivative of the sum is the sum of the derivatives.

26
Q

What is the integral of a constant k with respect to x?

A

kx + C

27
Q

True or False: The derivative of a product of two functions requires the use of the product rule.

A

True

28
Q

What is the limit definition of the derivative?

A

f’(x) = lim(h -> 0) [(f(x + h) - f(x))/h]

29
Q

What does the term ‘critical point’ refer to?

A

A critical point is where the derivative is zero or undefined.

30
Q

What is the integral of (1 + x^2) with respect to x?

A

x + (1/3)x^3 + C

31
Q

What is the derivative of sin(x^2)?

A

2x cos(x^2)

32
Q

Fill in the blank: A function is ________ if it has a derivative at every point in its domain.

A

differentiable

33
Q

What is the integral of tan(x) with respect to x?

A

-ln|cos(x)| + C

34
Q

True or False: The limit of a function at a point can be found using substitution.

A

True

35
Q

What is the derivative of cos(x)?

A

-sin(x)

36
Q

What is the integral of a constant multiplied by a function?

A

k * ∫f(x)dx = k * F(x) + C

37
Q

What is the difference between definite and indefinite integrals?

A

Definite integrals have limits and yield a numerical value, while indefinite integrals do not have limits and yield a function.

38
Q

What is the derivative of ln(x^2)?

A

2/x

39
Q

What is the integral of (ax + b)^n with respect to x?

A

(1/(a(n+1)))(ax + b)^(n+1) + C

40
Q

True or False: The second derivative of a function gives information about its concavity.

A

True

41
Q

What is the derivative of the inverse function f^-1(x)?

A

If y = f(x), then dy/dx = 1/(f’(f^-1(y)))

42
Q

What is the integral of sin^2(x) with respect to x?

A

(1/2)(x - sin(x)cos(x)) + C

43
Q

Fill in the blank: A function is ________ at a point if it can be drawn without lifting the pencil.

A

continuous

44
Q

What is the derivative of the exponential function a^x?

A

a^x ln(a)

45
Q

What is the integral of cos^2(x) with respect to x?

A

(1/2)(x + sin(x)cos(x)) + C

46
Q

Derivatives of arctan, arcsin, arccos.

A

(a/a^2+x^2) for arctan, 1/root(a^2-x^2) for arcsin, -1/root(a^2-x^2) for arccos

47
Q

Derivatives of arctan, arcsin, arccos.

A

(a/a^2+x^2) for arctan, 1/root(a^2-x^2) for arcsin, -1/root(a^2-x^2) for arccos

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