Module 3: Differentiation & its Applications

Flashcards based on lecture notes about Differentiation and its Applications, Standard Trig functions, Derivatives, Reciprocal Trig functions, Hyperbolic functions, Inverse Trig Functions and Related Rates

What is the domain of y = cos(x)?

The domain of cos(x) is R (all real numbers)

What is the range of y = cos(x)?

The range of cos(x) is [-1, 1].

What is the period of y = cos(x)?

The period of cos(x) is 2π.

What is the domain of y = sin(x)?

The domain of sin(x) is R (all real numbers).

What is the range of y = sin(x)?

The range of sin(x) is [-1, 1].

What is the period of y = sin(x)?

The period of sin(x) is 2π.

sin(x) =

cos(x - π/2).

cos(x) =

sin(x + π/2).

What is the domain of tan(x)?

{x ∈ R : x ≠ π/2 + nπ, n ∈ Z}.

What is the range of tan(x)?

R (all real numbers).

What is the period of tan(x)?

π.

d/dx (sin(x)) =

cos(x).

d/dx (cos(x)) =

-sin(x).

sec(x) =

1/cos(x).

What is the period of sec(x)?

2π.

What is the range of sec(x)?

(-∞, -1] ∪ [1, ∞).

What is the domain of sec(x)?

{x ∈ R : x ≠ (π/2) + nπ, n ∈ Z}.

cosec(x) =

1/sin(x) also written as csc(x).

What is the period of csc(x)?

2π.

What is the range of csc(x)?

(-∞, -1] ∪ [1, ∞).

What is the domain of csc(x)?

{x ∈ R : x ≠ nπ, n ∈ Z}.

cot(x) =

1/tan(x).

What is the period of cot(x)?

π.

What is the domain of cot(x)?

{x ∈ R : x ≠ nπ, n ∈ Z}.

What is the range of cot(x)?

R (all real numbers).

d/dx (tan(x)) =

sec²(x).

d/dx (sec(x)) =

tan(x)sec(x).

d/dx (cosec(x)) =

-cot(x)cosec(x).

d/dx (cot(x)) =

-cosec²(x).

sinh(x) =

(e^x - e^-x) / 2.

What is the domain of sinh(x)?

R (all real numbers).

What is the range of sinh(x)?

R (all real numbers).

cosh(x) =

(e^x + e^-x) / 2.

What is the domain of cosh(x)?

R (all real numbers).

What is the range of cosh(x)?

[1, ∞).

tanh(x) =

sinh(x) / cosh(x).

What is the domain of tanh(x)?

R (all real numbers).

What is the range of tanh(x)?

(-1, 1).

What is the hyperbolic identity analogue to cos²(x) + sin²(x) = 1?

cosh²(x) - sinh²(x) = 1.

d/dx (cosh(x)) =

sinh(x).

d/dx (sinh(x)) =

cosh(x).

d/dx (tanh(x)) =

sech²(x).

How do we define inverse trigonometric functions since trigonometric functions are periodic and not one-to-one?

Restricting the domain to an interval where the function is one-to-one (1-1).

To define the inverse sine function, y = sin⁻¹(x), what is the restricted domain of y = sin(x)?

[-π/2, π/2].

What is the domain of y = sin⁻¹(x)?

[-1, 1].

What is the range of y = sin⁻¹(x)?

[-π/2, π/2].

To define the inverse cosine function, y = cos⁻¹(x), what is the restricted domain of y = cos(x)?

[0, π].

What is the domain of y = cos⁻¹(x)?

[-1, 1].

What is the range of y = cos⁻¹(x)?

[0, π].

To define the inverse tangent function, y = tan⁻¹(x), what is the restricted domain of y = tan(x)?

(-π/2, π/2).

What is the domain of y = tan⁻¹(x)?

R (all real numbers).

What is the range of y = tan⁻¹(x)?

(-π/2, π/2).

d/dx (sin⁻¹(x)) =

1 / √(1 - x²).

d/dx (cos⁻¹(x)) =

-1 / √(1 - x²).

d/dx (tan⁻¹(x)) =

1 / (1 + x²).