Calculus: Differentiation, Integration and Taylor Series

• This unit covers calculus topics which include differentiation, integration, Taylor series, and calculus of variations.
• After completing the unit, you will know:
• How to differentiate and integrate functions of a single variable
• How to perform partial differentiation and multiple integrals for functions with multiple variables
• How to approximate a function in a Taylor series
• The basic concepts of calculus of variations

Calculus Introduction

• Functions express relationships between variables; y = f(x) formalizes with y as the dependent and x as the independent variable.
• Differentiation finds the rate at which a dependent variable changes with respect to an independent variable like velocity.
• Partial differentiation explores the rate of change with respect to each independent variable with multiple independent variables.
• Integration can undo differentiation and can be considered the anti-derivative.
• In calculus of variations, differentiation expands to functionals which are maps from functions to real numbers, maximizing or minimizing output.

Differentiation and Integration

• For a single variable, it begins with how a function changes with respect to its argument.
• Example is the velocity of a car, v(t), measures how fast it travels at time t.
• Average velocity, represents the rate at which a car travels over a time interval.
• v(t) = As/At, where As is the change in position.
• The time t is the argument of the function v, which relates covered distance and time.
• Considering a function f(x), fixing a value of x, and considering the function's value when changing, we can assume the function is continuous.
• Ax denotes the change in x, and x → x + ∆x represents x changing from x to x + ∆x.
• The idea of a derivative is an instantaneous rate of change.
• Gradient or first derivative of f is expressed as: f'(x) = df(x)/dx = lim (∆x→0) [f(x + ∆x) - f(x)] / ∆x.
• A function is differentiable at x only if the limit exists at x = x.
• Limit requires the quotient to approach the same value, f'(x), from both the left and right of x.

Example Derivative

• Derivative f'(x) can be the slope of the tangent line at a point x.
• f(x) = x² using the definition of derivatives:
• f'(x) = lim (∆x→0) [f(x + ∆x) - f(x)] / ∆x
• f'(x) = lim (∆x→0) [(x + ∆x)² - x²] / ∆x
• f'(x) = lim (∆x→0) [x² + 2x∆x + (∆x)² - x²] / ∆x
• f'(x) = lim (∆x→0) [2x∆x + (∆x)²] / ∆x
• f'(x) = lim (∆x→0) ∆x(2x + ∆x) / ∆x
• f'(x) = lim (∆x→0) (2x + ∆x)
• f'(x) = 2x
• Where ∆x becomes infinitely small as it approaches zero, can be canceled from numerator/denominator.
• For differentiability at x, a function must be continuous, but continuity doesn't guarantee differentiability shown by examining graphs.
• The derivative (slope) is -1, but if approaching x = 0 from the right, the limit of the quotient [f(x + ∆x) - f(x)] / ∆x is +1 .
• Limits disagree, derivative of f(x) = |x| isn't defined at x = 0.
• Definition and laws of limits can derive fundamental functions:
• d/dx xⁿ = nxⁿ⁻¹
• d/dx eᵃˣ = aeᵃˣ
• d/dx ln(ax) = 1/x
• d/dx sin(ax) = a cos(ax)
• d/dx cos(ax) = -a sin(ax)
• d/dx tan(ax) = a / cos²(ax)

Higher Order Derivatives

• Derivatives are themselves functions, derivatives can be considered of derivatives to denote higher order, the same way.
• For the second derivative: f''(x) = d f'(x) / dx = lim (∆x→0) [f'(x + ∆x) - f'(x)] / ∆x.
• The nth derivative is written as: f(ⁿ)(x) =df⁽ⁿ⁻¹⁾ / dx = lim (∆x→0) [f⁽ⁿ⁻¹⁾(x + ∆x) - f⁽ⁿ⁻¹⁾(x)] / ∆x.

Stationary points

• At x = 0, a local minimum is established (point 0,0), the function on sides greater.

• Graph tangent at point is horizontal, its equal zero.

• Slope of the line tangent to f at x = 0 (derivative f'(0)), has the same point.

• Points where the derivatives equal zero are "stationary".

• Illustrated by a previous graphic, tangent is undefined at points.

• For f(x) = |x|, (0, 0) is a critical point with zero derivative.

• Different stationary points:

• Function f has a max. at stationary point x = a if f'(a) = 0 and f''(a) < 0.

• The function f has a min. at stationary point x = a if f'(a) = 0 and f''(a) > 0.

• A point x = a is "saddle point" if f'(a) = 0 and f" changes sign.

• Maximums/Minimums found this way might not be global. Only point extrema.

Rules of Differentiation

• Functions with a constant and variable part, like f(x) = a * g(x), where a is constant and g depends on variable x.
• Derivative is: f’(x) = a * d/dx g(x) = a* g'(x)
• Functions products of two written as f(x) = u(x)v(x).

Product Rule

• Differentiating functions that can be expressed as products, the method for finding the derivative relies knowing how to differentiate u/v.
• Example: (g(x) =x² sin(x)):
• Decomposed as g(x) = u(x) * v(x) where (u(x)=x²) and (v(x)=sin(x)).
• Function can decompose, choose easy differentiation than f.
• Calculating a derivatives of a product, apply method to create easier calculations than using equation 1.2's definition.
• Product rule obtained (equation 1.2) as:
• f(x + ∆x) - f(x) = u(x + ∆x)v(x + ∆x) – u(x)v(x)
• = u(x + ∆x) [v(x + ∆x) – v(x)] + v(x)[u(x + ∆x) – u(x)].
• added/subtracted v(x)u(x+∆x) to factor, inserting into definition of this derivative:
• d/dx = lim (∆x→0) {u(x + ∆x) [v(x + ∆x) - v(x)] + v(x)[u(x + ∆x) - u(x)]} / ∆x
• As ∆x approaches zero and terms are derivatives:
• Product rule equation = f' = d/dx [u(x)v(x)] = u(x)(dv(x)/dx)) + v(x) (du(x)/dx) = uv' + vu'.
• Derivatives of three+ functions:
• Given f(x) = u(x)v(x)w(x)
• f'(x) = d/dx = u'(vw) + vw * du = uv'w + uw'v+vw'.
• Example, with u(x)=x² and v(x)=sin
• d/dx x²sin(x) = u dv/dx + sin(x) d(x²)/dx
• derivative = x²cos(x) + 2xsin(x).

Chain Rule

• Writing functions as composition in function, example f(x)=(x-1)² as f(x)=u²(x) with u(x) = x - 1.
• The idea is differentiating outer function to u getting f'(u), inner function alone.
• Derivative of the inner to respect to x in order to get u'(x)
• Combined the values for df / dx = (df / du) * (du / dx).

Chain Rule Example

• Used for functions derivative, extend to functions to functions.
• Derivative using repeated, function of functions chains:
• f(x) = (x - 1)².
• We can rewrite: f(x)= u² (x), u(x) = (x - 1)
• Derivative calculation as: df / dx = (df / du) * (du / dx)
• =2u * 1
• =2u and dx / dx=1
• = 2*(x-1)
• Use to calculate functions with f(x)=1/v(x) to express the -1 power then apply.
• df / dx = (df / dv) * (dv / dx)
• =-v⁻²dv / dx
• 1/v² (x) *dv/ dx
• Derivatives for quotients when combined with product and chain.
• For functions such as (f(x) = u(x)/v(x)):
• f(x) = u(x)[1/v(x)] to get
• d / dx f(x) = u(x (d/dx) (1 /v(x )= u(x) (-dv(x)/dx + du(x)/dx
• To gain d/dx (1/ v(x)), obtain f' = (vu' - uv') / v².

Integrals functions

• First focus rates a single variable using the derivative.
• Returning traveling, average velocity is as distance over, v= ds/ dt
• Considering is for instanteous velocity of d / dt.
• Derivatives relative to position equal instantaneous
• Calculate an estimated value of the car over delta.
• Constant v = vDt, of position. Area of the constant is the position.
• The velocity isn't consistant for a calculation to the value.
• Look intervals of intervals to see where the rectangles have, the the car over short lengths.

Area for the Functions

• Function to evaluate the height to an approximation.
• Error in this calculation, intervals reduces. To the total contributions of a interval.
• Divide the small limits which is defined in x variable that is on the intervals of the approches above to the of area with = d/dx

Closed functions

• Approches. function with integrable by whether the function is continous

Evaluating integrals

• The derivative is the determining to of the of a limit or to determining of a limit to rectangles.

Integral and Derivtation

• Of the a, determining to determine the integrals over time is by of formula: d =x dx is x b b of over integrals.

Integration Equations

• Equations of integrals are shown along of the equations of of a b in a set.

Logrithmic evaluation

• Intergrals as which: x dx - dx (n )+

Integrals (Substituion)

• Intergarals as substition from what is from the: f xdx - f dx

Integral (Integration)

• Integrals (power to to what the can be can to as intergratating, from what is to it power

Equation (Derivatites)

• Can to be written in can be a that which from what integral
• The a the to by
• From integrals is not limited functions to is in b, so functions to evaluate as a The that function is to be:

Approaching as intergals

• Approches to the that limit formula x - x of to in to from b.
• Functions the and b which formula that from function.

Multi-variable equation

• General of the of vector to 10x that can what the intergates
• F in what and to