5.1 Introduction to Differential Calculus

Tags

Differentiation

Limits

First principles

Asymptotes

Convergent

Divergent

Continuous

Implicit differentiation

Chain rule

Product rule

Quotient rule

Trigonometric functions

Exponential and Logarithmic functions

f(x)

is as close as we like to some real number

A

for all

x

sufficiently close to (but not equal to)

a

, then we say that

f(x)

has a limit of

A

x

approaches

a

, and we write

\lim\limits_{x \to a} f(x) = A

In this case,

f(x)

is said to converge to

A

x

approaches

a

. Otherwise, we say a limit is divergent.

Figure 5.1.1 Convergence of limits for continuous (left) and discontinuous function (right)

We define continuity using limits. Consider a real function

f

defined on an open interval containing the value

a

. We say

f

is continuous at

x = a

\lim\limits_{x \to a} f(x) = f(a)

f

is continuous at

x = a

for all

a \epsilon ℝ

, we say

f

is continuous on

ℝ

There are three different methods to solve limits.

Substitution

00\frac0000​ , ∞−∞∞ - ∞∞−∞, 0−∞0 - ∞0−∞ form: factorization

∞∞\frac ∞∞∞∞​ form: divide by the highest power

We can identify asymptotes using limits.

Always investigate

\rightarrow ∞

\rightarrow -∞

and possible points of singularity

x = a

If lim⁡x→±∞f(x)=L\lim\limits_{x\rightarrow ± ∞} f(x) = Lx→±∞lim​f(x)=L we have a horizontal asymptote: y=Ly = Ly=L

If lim⁡x→a+f(x)=∞\lim\limits_{x \rightarrow a+}f(x) =∞x→a+lim​f(x)=∞ and lim⁡x→a−f(x)=−∞\lim\limits_{x \rightarrow a-}f(x) = -∞x→a−lim​f(x)=−∞, we have a vertical asymptote: x=ax = ax=a

Point of singularity, in its simplest term denotes the point of discontinuity.

There is a squeeze theorem, used in Riemann sum later in integration.

g(x) ≤ f(x) ≤ h(x)

and

\lim\limits_{x\rightarrow a}g(x) = \lim\limits_{x\rightarrow a}h(x) = L

, then

\lim\limits_{x\rightarrow a} f(x) = L

You can also use L’Hopital’s rule if you have the form:

\frac00

\frac∞∞

, or

\frac 0∞

, and

\lim\limits_{x\rightarrow a}\frac{f'(x)}{g'(x)}

exists.

\lim\limits_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim\limits_{x\rightarrow a}\frac{f'(x)}{g'(x)}

*necessary condition:

f

and

g

are differentiable over the open interval

I

g'(x) ≠ 0

for all

x

I

, with

x ≠ c

Rate is a comparison between two quantities with different units.

Average rate of change over a particular time interval:

\frac{∆f}{∆x}=\frac{f(b) - f(a)}{b - a}=\frac{f(a+h) - f(a)}h

Differentiation, or Derivatives talks about the rate of change of a function with respect to a variable taken to its limit. Therefore, a derivative at a point has a geometrical interpretation - the gradient of a tangent at that point.

Figure 5.2.1 Graphical representation of differentiation from first principles

At a certain point x=a,x = a,x=a, f′(a)=lim⁡b→af(b)−f(a)b−a=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim\limits_{b\rightarrow a}\frac{f(b) - f(a)}{b - a}=\lim\limits_{h\rightarrow 0}\frac{f(a + h) - f(a)}hf′(a)=b→alim​b−af(b)−f(a)​=h→0lim​hf(a+h)−f(a)​

General point xxx, dfdx=f′(x)=lim⁡x→af(x)−f(a)−a=lim⁡h→0f(x+h)−f(x)h\frac{df}{dx}=f'(x) =\lim\limits_{x\rightarrow a} \frac{f(x) - f(a)}{- a}=\lim\limits_{h\rightarrow 0} \frac{f(x + h) - f(x)}{h}dxdf​=f′(x)=x→alim​−af(x)−f(a)​=h→0lim​hf(x+h)−f(x)​

This is finding the derivatives using the newton quotient, or the first principle. We call this ‘differentiating from the first principles’ and can generalize some of the derivative functions into formulas.

Differentiation is a linear operation, and thus preserves vector addition and scalar multiplication.

ddx[f(x)+g(x)]=f′(x)+g′(x)\frac d{dx}[f(x) + g(x)] = f'(x) + g'(x)dxd​[f(x)+g(x)]=f′(x)+g′(x)

ddxcf(x)=cf′(x)\frac d{dx}cf(x) = cf'(x)dxd​cf(x)=cf′(x)

Rules:

Chain Rule: dfdx=dfdt⋅dtdx, ddxf(g(x))=f′(g(x))g′(x)\frac{df}{dx}=\frac{df}{dt}\cdot \frac{dt}{dx}, ~\frac d{dx}f(g(x)) = f'(g(x))g'(x)dxdf​=dtdf​⋅dxdt​, dxd​f(g(x))=f′(g(x))g′(x)

Product Rule: ddxf(x)g(x)=f′(x)g(x)+f(x)g′(x)\frac d{dx}f(x)g(x) = f'(x)g(x) + f(x)g'(x)dxd​f(x)g(x)=f′(x)g(x)+f(x)g′(x)

Quotient Rule: ddxf(x)g(x)=f′(x)g(x)−g′(x)f(x)[g(x]2\frac d{dx}\frac{f(x)}{g(x)} = \frac{f'(x)g(x) - g'(x)f(x)}{[g(x]^2}dxd​g(x)f(x)​=[g(x]2f′(x)g(x)−g′(x)f(x)​.

Apply these rules with different derivative formulas in the formula booklet accordingly.

5.1 Introduction to differential calculus

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5.1 Introduction to Differentiation