This chapter introduces fundamental concepts of calculus, focusing on limits and derivatives, which are essential for understanding changes in functions.
Limits and Derivatives – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Limits and Derivatives chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
lim (x -> a) f(x) = L
The limit of f(x) as x approaches a is L. This indicates the behavior of f(x) as x gets infinitely close to a.
f'(x) = lim (h -> 0) [(f(x+h) - f(x))/h]
The derivative of f at point x, denoted f'(x), quantifies the rate of change of f with respect to x.
s = 4.9t²
s is the distance in meters covered by a freely falling body after t seconds, derived from kinematic equations.
Average velocity = (s(t2) - s(t1)) / (t2 - t1)
This formula calculates the average velocity between two time intervals, useful in approximating instantaneous velocity.
lim (x -> a) [f(x) + g(x)] = lim (x -> a) f(x) + lim (x -> a) g(x)
For functions f and g, the limit of their sum equals the sum of their limits.
lim (x -> a) [f(x) * g(x)] = lim (x -> a) f(x) * lim (x -> a) g(x)
The limit of the product of two functions is the product of their limits.
lim (x -> a) [f(x) / g(x)] = lim (x -> a) f(x) / lim (x -> a) g(x) (if g(a) ≠ 0)
The limit of the quotient of two functions is the quotient of their limits when the denominator is not zero.
d/dx (c * f(x)) = c * f'(x)
The derivative of a constant multiplied by a function is the constant multiplied by the derivative of that function.
d/dx [f(x) + g(x)] = f'(x) + g'(x)
The derivative of the sum of two functions is the sum of their derivatives.
d/dx (x^n) = n*x^(n-1)
The derivative of x raised to the power of n is n multiplied by x raised to the power of n-1.
Equations
lim (x -> 2) 4x^2 - 2x + 3 = 16 - 4 + 3 = 15
Example of finding the limit of a polynomial function as x approaches 2.
lim (x -> 0) (sin x)/x = 1
A standard limit that establishes the behavior of the sine function near zero.
lim (x -> 0) (1 - cos x)/x^2 = 0.5
Another standard limit representing the behavior of the cosine function near zero.
f'(x) = 3x^2 + 2
Derivative of the polynomial function f(x) = x^3 + 2x + C.
f'(a) = lim (x -> a) [(f(x) - f(a)) / (x - a)]
This is the definition of the derivative using limits.
d/dx (sin x) = cos x
The derivative of the sine function.
d/dx (cos x) = -sin x
The derivative of the cosine function.
d/dx (tan x) = sec^2 x
The derivative of the tangent function.
s = ut + (1/2)at²
Equation of motion which helps derive limits and derivatives in kinematics.
d/dx (f(x) * g(x)) = f'(x)g(x) + g'(x)f(x)
Product rule for derivatives, showing how to differentiate products of functions.
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