Канал: Matheracy Matters

Indefinite Integral of 1/(1-x²)

Indefinite Integral of 1/(1-x²)

Integration by Substitution I

Integration by Substitution I

Indefinite Integral: an Example (5)

Indefinite Integral: an Example (5)

Indefinite Integral: an Example (4)

Indefinite Integral: an Example (4)

Indefinite Integral: an Example (3)

Indefinite Integral: an Example (3)

Indefinite Integral: an Example (2)

Indefinite Integral: an Example (2)

Indefinite Integral of 1/(1+x²)

Indefinite Integral of 1/(1+x²)

Basic Integration Rules VI

Basic Integration Rules VI

Basic Integration Rules V

Basic Integration Rules V

Basic Integration Rules IV

Basic Integration Rules IV

Indefinite Integral: an Example (1)

Indefinite Integral: an Example (1)

Basic Integration Rules III

Basic Integration Rules III

Basic Integration Rules II

Basic Integration Rules II

Basic Integration Rules I

Basic Integration Rules I

Projectile Motion IV: Trajectory Equation

Projectile Motion IV: Trajectory Equation

Projectile Motion III: Maximum Height

Projectile Motion III: Maximum Height

Projectile Motion II: Vertical Component

Projectile Motion II: Vertical Component

Projectile Motion I: Horizontal Component

Projectile Motion I: Horizontal Component

Motion in a Plane II

Motion in a Plane II

Motion in a Plane I

Motion in a Plane I

Expanding Speed of Level Surface

Expanding Speed of Level Surface

Rising Speed of Surface Level

Rising Speed of Surface Level

Equations of Motion for Constant Acceleration III

Equations of Motion for Constant Acceleration III

Volume of a Truncated Cone

Volume of a Truncated Cone

Equations of Motion for Constant Acceleration II

Equations of Motion for Constant Acceleration II

Equations of Motion for Constant Acceleration I

Equations of Motion for Constant Acceleration I

Velocity of Angle: an Example

Velocity of Angle: an Example

Definition of Indefinite Integral

Definition of Indefinite Integral

Regarding Moving Points: an Example

Regarding Moving Points: an Example

Velocity and Acceleration II

Velocity and Acceleration II

Velocity and Acceleration I

Velocity and Acceleration I

Newton-Raphson Method: Approximating f(x)=0

Newton-Raphson Method: Approximating f(x)=0

Original Newton's Method: Approximating x³-2x-5=0

Original Newton's Method: Approximating x³-2x-5=0

Maximizing Cross Area A ⟹ θ, MaxA=? [3rd Method]

Maximizing Cross Area A ⟹ θ, MaxA=? [3rd Method]

Maximizing Cross Area A ⟹ θ, MaxA=? [2nd Method]

Maximizing Cross Area A ⟹ θ, MaxA=? [2nd Method]

Maximizing Cross Area A ⟹ x, y, MaxA=? [1st Method]

Maximizing Cross Area A ⟹ x, y, MaxA=? [1st Method]

Implicit Differentiation: an Examle

Implicit Differentiation: an Examle

Maximizing Water Area of a Canal

Maximizing Water Area of a Canal

Power Series: 1/5+2²/5²+3²/5³+4²/5⁴+···=?

Power Series: 1/5+2²/5²+3²/5³+4²/5⁴+···=?

Power Series: 1/5+2/5²+3/5³+4/5⁴+···=?

Power Series: 1/5+2/5²+3/5³+4/5⁴+···=?

Proving Inequality: (aⁿ+bⁿ)/2 ≧ [(a+b)/2]ⁿ [2nd & 3rd Method]

Proving Inequality: (aⁿ+bⁿ)/2 ≧ [(a+b)/2]ⁿ [2nd & 3rd Method]

Proving Inequality: (aⁿ+bⁿ)/2 ≧ [(a+b)/2]ⁿ [1st Method]

Proving Inequality: (aⁿ+bⁿ)/2 ≧ [(a+b)/2]ⁿ [1st Method]

A Glimpse on Jensen's Inequality

A Glimpse on Jensen's Inequality

Maximizing Volume of a Cone

Maximizing Volume of a Cone

Shortest Distance from a Point to a Curve

Shortest Distance from a Point to a Curve

Maximum Area of the Triangle Formed by the Tangent of a Curve

Maximum Area of the Triangle Formed by the Tangent of a Curve

Maximum Volume of the Right Circular Cone Circumscribed by a Sphere of Radius r

Maximum Volume of the Right Circular Cone Circumscribed by a Sphere of Radius r

Minimum Volume of the Right Circular Cone that Circumscribes a Sphere of Radius r

Minimum Volume of the Right Circular Cone that Circumscribes a Sphere of Radius r

Minimum Volume of a Box

Minimum Volume of a Box

Graph of a Cubic Function (3)

Graph of a Cubic Function (3)

Point Symmetry of Graph of Cubic Function (2)

Point Symmetry of Graph of Cubic Function (2)

Point Symmetry of Graph of Cubic Function (1)

Point Symmetry of Graph of Cubic Function (1)

Point Symmetry of a Function

Point Symmetry of a Function

Derivative Test: Example (2) [2nd Method]

Derivative Test: Example (2) [2nd Method]

Inflection Point: Example (2)

Inflection Point: Example (2)

Inflection Point: Example (1)

Inflection Point: Example (1)

Derivative Test: Example (1) [2nd Method]

Derivative Test: Example (1) [2nd Method]

The Second Derivative Test for a Relative (Local) Extrema

The Second Derivative Test for a Relative (Local) Extrema

Inflection Point of a Function

Inflection Point of a Function

Finding Global (Absolute) Minimum: Example

Finding Global (Absolute) Minimum: Example

Concavity Test

Concavity Test

Derivative Test: Example (2) [1st Method]

Derivative Test: Example (2) [1st Method]

Definition of Concavity

Definition of Concavity

Derivative Test: Example (1) [1st Method]

Derivative Test: Example (1) [1st Method]

The First Derivative Test for a Relative (Local) Minimum

The First Derivative Test for a Relative (Local) Minimum

Local Extrema: Example

Local Extrema: Example

The First Derivative Test for a Relative (Local) Maximum

The First Derivative Test for a Relative (Local) Maximum

Gradient of the Cycloid Curve at Point P(x,y)=?

Gradient of the Cycloid Curve at Point P(x,y)=?

Test for Increasing and Decreasing Functions

Test for Increasing and Decreasing Functions

Definitions of Increasing and Decreasing Functions

Definitions of Increasing and Decreasing Functions

Mean Value Theorem: an Example

Mean Value Theorem: an Example

Mean Value Theorem: an Example (2)

Mean Value Theorem: an Example (2)

Mean Value Theorem: an Example

Mean Value Theorem: an Example

Mean Value Theorem: an Example (1)

Mean Value Theorem: an Example (1)

Mean Value Theorem

Mean Value Theorem

Rolle's Theorem: an Example

Rolle's Theorem: an Example

Rolle's Theorem

Rolle's Theorem

Higher Order Derivatives: an Exmaple

Higher Order Derivatives: an Exmaple

Higher Order Derivatives of 1/(x-k)

Higher Order Derivatives of 1/(x-k)

Higher Order Derivatives: an Exmaple

Higher Order Derivatives: an Exmaple

Definition of Relative Extrema, Definition of Critical Number

Definition of Relative Extrema, Definition of Critical Number

y=x+3, y=x-1 are tangent to y=x³+ax+b ⟹ a, b=?

y=x+3, y=x-1 are tangent to y=x³+ax+b ⟹ a, b=?

Definition of Extrema, Extreme Value Theorem

Definition of Extrema, Extreme Value Theorem

A(p, eᵖ), B(p+1, eᵖ⁺¹), AB: Tangent Line to y=eˣ⁺ᵃ at x=t ⟹ a=?

A(p, eᵖ), B(p+1, eᵖ⁺¹), AB: Tangent Line to y=eˣ⁺ᵃ at x=t ⟹ a=?

Parametric Differentiation: Tangent Line of a Curve

Parametric Differentiation: Tangent Line of a Curve

Implicit Differentiation: an Example

Implicit Differentiation: an Example

Normal Line to a Curve: an Example

Normal Line to a Curve: an Example

f(x)=2xe⁻³ˣ, ∀x∈ℝ: f''(x)+af'(x)+bf(x)=0 ⟹ a,b=?

f(x)=2xe⁻³ˣ, ∀x∈ℝ: f''(x)+af'(x)+bf(x)=0 ⟹ a,b=?

Slope of Tangent Line to a Curve

Slope of Tangent Line to a Curve

Normal Line to Tangent Curves: an Example

Normal Line to Tangent Curves: an Example

Tangent Line to a Curve from a Point: an Example

Tangent Line to a Curve from a Point: an Example

f(x)=eᵃˣsinbx (b≠0), ∀x∈ℝ: f''(x)-2f'(x)+3f(x)=0 ⟹ a, b=?

f(x)=eᵃˣsinbx (b≠0), ∀x∈ℝ: f''(x)-2f'(x)+3f(x)=0 ⟹ a, b=?

Equation of Tangent Line to a Circle

Equation of Tangent Line to a Circle

f(θ)=cosθ-sinθ, f''(θ)=0 ⟹ θ∈[0,π/2]=?

f(θ)=cosθ-sinθ, f''(θ)=0 ⟹ θ∈[0,π/2]=?

y=bx+c is tangent to y=ax² and y=lnx ⟹ a+b+c=?

y=bx+c is tangent to y=ax² and y=lnx ⟹ a+b+c=?

f(x)=ⁿ∑ₖ₌₁eᵏˣ, lim(x→0)(1/x)[ln{f(x)/n}]=6 ⟹ n∈ℕ=?

f(x)=ⁿ∑ₖ₌₁eᵏˣ, lim(x→0)(1/x)[ln{f(x)/n}]=6 ⟹ n∈ℕ=?

f(x)=∛(x²-1)/(x²+1)⟹f'(x)=?

f(x)=∛(x²-1)/(x²+1)⟹f'(x)=?

f(x)=aˣ ⟹ f'(x)=?

f(x)=aˣ ⟹ f'(x)=?

Power Series Expansion of aˣ [1st Method]

Power Series Expansion of aˣ [1st Method]

1²+2²x+3²x²+4²x³+···=?

1²+2²x+3²x²+4²x³+···=?

lim{x→0}sinx/x=? [2nd Method]

lim{x→0}sinx/x=? [2nd Method]

Derivative of Cosine Function [2nd Method]

Derivative of Cosine Function [2nd Method]

Power Series Expansion of 4/(1-x)(1+3x)

Power Series Expansion of 4/(1-x)(1+3x)

Derivative of Sine Function [2nd Method]

Derivative of Sine Function [2nd Method]

Power Series Expasion of 1/√(1-x²)

Power Series Expasion of 1/√(1-x²)

Power Series of cosx & sinx [2nd Method]

Power Series of cosx & sinx [2nd Method]

Logarithmic Series ln(1+x)=x-x²/2+x³/3-x⁴/4+···

Logarithmic Series ln(1+x)=x-x²/2+x³/3-x⁴/4+···

Polynomial Approximation of ln(1+x)

Polynomial Approximation of ln(1+x)

Polynomial Approximation of cosx

Polynomial Approximation of cosx

Geometric Series 1/(1+x)=1-x+x²-x³+x⁴-···

Geometric Series 1/(1+x)=1-x+x²-x³+x⁴-···

Maclaurin Series of cosx

Maclaurin Series of cosx

Polynomial Approximation of sinx

Polynomial Approximation of sinx

Maclaurin Series of sinx

Maclaurin Series of sinx

Polynomial Approximation of eˣ

Polynomial Approximation of eˣ

Maclaurin Series of eˣ

Maclaurin Series of eˣ

Maclaurin Series of (1+x)ⁿ & Newton's Generalized Binomial Series

Maclaurin Series of (1+x)ⁿ & Newton's Generalized Binomial Series

Maclaurin Series

Maclaurin Series

Binomial Expansion of (1+x)ⁿ, n∈ℤ

Binomial Expansion of (1+x)ⁿ, n∈ℤ

f(x)=tanx ⟹ f'(x)=?

f(x)=tanx ⟹ f'(x)=?

f(x)=sinx ⟹ f'(x)=? [1st & 2nd Method]

f(x)=sinx ⟹ f'(x)=? [1st & 2nd Method]

f(x)=cosx ⟹ f'(x)=?

f(x)=cosx ⟹ f'(x)=?

Quotient Rule of Differentiation

Quotient Rule of Differentiation

Intermediate Value Theorem

Intermediate Value Theorem

Product Rule of Differentiation

Product Rule of Differentiation

1²/2¹+2²/2²+3²/2³+4²/2⁴+···=?

1²/2¹+2²/2²+3²/2³+4²/2⁴+···=?

Chain Rule of Differentiation

Chain Rule of Differentiation

Sum & Difference of Differentiation

Sum & Difference of Differentiation

1+3/2+5/4+7/8+9/16+···=? [2nd Method]

1+3/2+5/4+7/8+9/16+···=? [2nd Method]

f(x)=eˣ ⟹ f'(x)=? [3rd & 4th Method]

f(x)=eˣ ⟹ f'(x)=? [3rd & 4th Method]

f(x)=xⁿ ⟹ f'(x)=? [3rd Method]

f(x)=xⁿ ⟹ f'(x)=? [3rd Method]

f(x)=eˣ ⟹ f'(x)=? [1st & 2nd Method]

f(x)=eˣ ⟹ f'(x)=? [1st & 2nd Method]

y=lnx ⟹ y'=? [2nd Method]

y=lnx ⟹ y'=? [2nd Method]

f(x)=lnx ⟹ f'(x)=? [1st Method]

f(x)=lnx ⟹ f'(x)=? [1st Method]

f(x)=xⁿ ⟹ f'(x)=? [1st & 2nd Method]

f(x)=xⁿ ⟹ f'(x)=? [1st & 2nd Method]

Basic Defintions of Differentiation: Derivative of a Function and Differentiability at a Point

Basic Defintions of Differentiation: Derivative of a Function and Differentiability at a Point

f(x〈0)=(e³ˣ-1)/tanx, f(x≥0)=-x²+4x+a, f(x): continuous at x=0 ⟹ a=?

f(x〈0)=(e³ˣ-1)/tanx, f(x≥0)=-x²+4x+a, f(x): continuous at x=0 ⟹ a=?

Definition of Continuity on a Closed Interval

Definition of Continuity on a Closed Interval

Definition of Continuity on an Open Interval

Definition of Continuity on an Open Interval

Definition of Continuity on a Point

Definition of Continuity on a Point

Radius of Convergence of a Power Series

Radius of Convergence of a Power Series

d'Alembert's Ratio Test: Extension

d'Alembert's Ratio Test: Extension

d'Alembert's Ratio Test: Examples

d'Alembert's Ratio Test: Examples

d'Alembert's Ratio Test

d'Alembert's Ratio Test

lim(x→π)√(1+acosx-b)/(x-π)² ⟹ a+b=?

lim(x→π)√(1+acosx-b)/(x-π)² ⟹ a+b=?

lim(x→2)(x³+ax²+bx+c)/(x²-4x+4)=3 ⟹ a, b, c=? [1st Method]

lim(x→2)(x³+ax²+bx+c)/(x²-4x+4)=3 ⟹ a, b, c=? [1st Method]

lim(x→2)(x³+ax²+bx+c)/(x²-4x+4)=3 ⟹ a, b, c=? [2nd Method]

lim(x→2)(x³+ax²+bx+c)/(x²-4x+4)=3 ⟹ a, b, c=? [2nd Method]

aₖ=[k(n-k)!+(k+1)]/(k+1)(n-k)! ⟹ Σ(n=0~∞)aₙ=?

aₖ=[k(n-k)!+(k+1)]/(k+1)(n-k)! ⟹ Σ(n=0~∞)aₙ=?

lim(n→∞)2ⁿ/n!=?

lim(n→∞)2ⁿ/n!=?

lim(x→0)(cosax-cosbx)/x²=? (ab≠0)

lim(x→0)(cosax-cosbx)/x²=? (ab≠0)

Limit of a Rational Function

Limit of a Rational Function

Definition of Euler's Number e

Definition of Euler's Number e

lim(x→0)(aˣ-1)/x=?

lim(x→0)(aˣ-1)/x=?

lim(x→0)[sin(x+h)-sinx]/[√(x+h)-√x]=?

lim(x→0)[sin(x+h)-sinx]/[√(x+h)-√x]=?

lim(x→0)x²cos(π/x)=?

lim(x→0)x²cos(π/x)=?

lim(x→0)sinx/x=? [1st Method]

lim(x→0)sinx/x=? [1st Method]

Squeeze Theorem: a Proof

Squeeze Theorem: a Proof

Squeeze (Sandwich) Theorem

Squeeze (Sandwich) Theorem

Geometric Series

Geometric Series

aₖ=lim(n→∞)[(8/k)ⁿ⁺¹]/[(8/k)ⁿ+1] ⟹ a₁+2a₂+3a₃+···+100a₁₀₀=?

aₖ=lim(n→∞)[(8/k)ⁿ⁺¹]/[(8/k)ⁿ+1] ⟹ a₁+2a₂+3a₃+···+100a₁₀₀=?

log(1+1/2²)+log(1+1/2⁴)+log(1+1/2⁸)++log(1+1/2¹⁶)+···=?

log(1+1/2²)+log(1+1/2⁴)+log(1+1/2⁸)++log(1+1/2¹⁶)+···=?

Definition of Limit (2)

Definition of Limit (2)

x!=x³-x ⟹ x=?

x!=x³-x ⟹ x=?

3/1·2·3+3/2·3·4+3/3·4·5+···=?

3/1·2·3+3/2·3·4+3/3·4·5+···=?

Definition of Limit (1)

Definition of Limit (1)

1+3/2+5/4+7/8+9/16+···=? [1st Method]

1+3/2+5/4+7/8+9/16+···=? [1st Method]

f(x)=(2x+1)/[√(x²-1)+2] ⟹ lim(x→∞)f(x)-lim(x→-∞)f(x)=?

f(x)=(2x+1)/[√(x²-1)+2] ⟹ lim(x→∞)f(x)-lim(x→-∞)f(x)=?

An Example of Limit

An Example of Limit

f(x)=[1/√3-1/√(x+3)]/x ⟹ lim(x→0)f(x)=?

f(x)=[1/√3-1/√(x+3)]/x ⟹ lim(x→0)f(x)=?

a₁/0!+a₂/1!+a₃/2!+···+aₙ/(n-1)!=1/(n+2)! ⟹ lim(n→∞)(6a₁-n²aₙ)=?

a₁/0!+a₂/1!+a₃/2!+···+aₙ/(n-1)!=1/(n+2)! ⟹ lim(n→∞)(6a₁-n²aₙ)=?

a₁=5, aₙ₊₁-aₙ=3, b₁=⅓, 1/b₁+1/b₂+···+1/bₙ=(n+1)² ⟹ lim(n→∞)aₙbₙ=?

a₁=5, aₙ₊₁-aₙ=3, b₁=⅓, 1/b₁+1/b₂+···+1/bₙ=(n+1)² ⟹ lim(n→∞)aₙbₙ=?

aₙ=√(an²+n)-√(an²-an), lim(n→∞)aₙ=5/3 ⟹ a=?

aₙ=√(an²+n)-√(an²-an), lim(n→∞)aₙ=5/3 ⟹ a=?

a+b+c=0 ⟹ (a²+b²+c²)/(b²-ac)=?

a+b+c=0 ⟹ (a²+b²+c²)/(b²-ac)=?

{x|(x+1)²+(x+2)³+(x+3)⁴=2}={α, β, γ, δ} ⟹ αβγδ=?

{x|(x+1)²+(x+2)³+(x+3)⁴=2}={α, β, γ, δ} ⟹ αβγδ=?

a₁=b₁=7, aₙ₊₁=4aₙ+3bₙ, bₙ₊₁=3bₙ+2aₙ ⟹ lim{n→∞}aₙ/bₙ=?

a₁=b₁=7, aₙ₊₁=4aₙ+3bₙ, bₙ₊₁=3bₙ+2aₙ ⟹ lim{n→∞}aₙ/bₙ=?

a₁=2/3, aₙ(3Sₙ-2)=3Sₙ² ⟹ aₙ=?

a₁=2/3, aₙ(3Sₙ-2)=3Sₙ² ⟹ aₙ=?

(1²+1)1!+(2²+1)2!+(3²+1)3!+···+(n²+1)n!=3600 ⟹ n=?

(1²+1)1!+(2²+1)2!+(3²+1)3!+···+(n²+1)n!=3600 ⟹ n=?

a₁=2, aₙ₊₁=aₙ(aₙ+2) ⟹ aₙ=?

a₁=2, aₙ₊₁=aₙ(aₙ+2) ⟹ aₙ=?

a₁=4, aₙ₊₁=aₙ(aₙ-2)+2 ⟹ aₙ=?

a₁=4, aₙ₊₁=aₙ(aₙ-2)+2 ⟹ aₙ=?

a₁=1, aₙ₊₁=aₙ/(aₙ+3) ⟹ aₙ=?

a₁=1, aₙ₊₁=aₙ/(aₙ+3) ⟹ aₙ=?

a₁=5, (aₙ₊₁-aₙ)/4=1+√(aₙ+4) ⟹ aₙ=?

a₁=5, (aₙ₊₁-aₙ)/4=1+√(aₙ+4) ⟹ aₙ=?

a₁=3, aₙ₊₁=3ⁿaₙ ⟹ aₙ=?

a₁=3, aₙ₊₁=3ⁿaₙ ⟹ aₙ=?

a₁=1, aₙ₊₁=3aₙ+2ⁿ ⟹ aₙ=?

a₁=1, aₙ₊₁=3aₙ+2ⁿ ⟹ aₙ=?

a₁=1, aₙ₊₁=3aₙ+n ⟹ aₙ=?

a₁=1, aₙ₊₁=3aₙ+n ⟹ aₙ=?

a₁=2, aₙ₊₁=n/(n+1)aₙ ⟹ aₙ=?

a₁=2, aₙ₊₁=n/(n+1)aₙ ⟹ aₙ=?

a₁=2, a₂=5, aₙ₊₂=4aₙ₊₁-3aₙ⟹ Sₙ=?

a₁=2, a₂=5, aₙ₊₂=4aₙ₊₁-3aₙ⟹ Sₙ=?

1·2+2·3+3·4+···+n(n+1)=?

1·2+2·3+3·4+···+n(n+1)=?

a₁=1, aₙ₊₁=aₙ+rⁿ ⟹ aₙ=? (r≠1)

a₁=1, aₙ₊₁=aₙ+rⁿ ⟹ aₙ=? (r≠1)

Recurrence Relation: Difference Sequence #sequence #recurrencerelation

Recurrence Relation: Difference Sequence #sequence #recurrencerelation

a₁=1, aₙ₊₁=aₙ+3n ⟹ aₙ=?

a₁=1, aₙ₊₁=aₙ+3n ⟹ aₙ=?

[(1-√5)/2]¹²=?

[(1-√5)/2]¹²=?

Sum of Geometric Sequence #geometricsequence #sigma

Sum of Geometric Sequence #geometricsequence #sigma

[(1+√5)/2]¹²=?

[(1+√5)/2]¹²=?

{Fₙ}={1, 1, 2, 3, 5, 8, 12, 21, ···} ⟹ Fₙ=?

{Fₙ}={1, 1, 2, 3, 5, 8, 12, 21, ···} ⟹ Fₙ=?

S(n)=|n-1|+|n-2|+···+|n-99| ⟹ minS(n)=?

S(n)=|n-1|+|n-2|+···+|n-99| ⟹ minS(n)=?

√(1+1/1²+1/2²)+√((1+1/2²+1/3²)+√((1+1/3²+1/4²)+‧‧‧+√((1+1/99²+1/100²)=?

√(1+1/1²+1/2²)+√((1+1/2²+1/3²)+√((1+1/3²+1/4²)+‧‧‧+√((1+1/99²+1/100²)=?

Sum of Arithmetic Sequence: an Example

Sum of Arithmetic Sequence: an Example

f(n)=(4n+√(4n²-1))/(√((2n+1)+√(2n-1)) ⟹ f(1)+f(2)+‧‧‧+f(60)=?

f(n)=(4n+√(4n²-1))/(√((2n+1)+√(2n-1)) ⟹ f(1)+f(2)+‧‧‧+f(60)=?

Sum of Arithmetic Sequence #arithmeticsequence #sigma

Sum of Arithmetic Sequence #arithmeticsequence #sigma

Geometric Sequence: an Example #geometricsequence

Geometric Sequence: an Example #geometricsequence

1²+2²+3²+···n²=? [4th Method]

1²+2²+3²+···n²=? [4th Method]