SYMKC Study

Flashcards covering derivatives, theorems, and applications of calculus.

What does lim (x→a-) f(x) represent?

The limit of f(x) as x approaches a from the left.

What does lim (x→a+) f(x) represent?

The limit of f(x) as x approaches a from the right.

How do you find critical points?

f'(x) = 0 or undefined

What is the condition for a function to be increasing?

f' >= 0

What is the condition for a function to be decreasing?

f' <= 0

What is the condition for a relative minimum?

f' changes from negative to positive.

What is the condition for a relative maximum?

f' changes from positive to negative.

How do you find absolute extrema?

Check endpoints and critical points.

What is the condition for concave up?

f'' > 0 or f' is increasing

What is the condition for concave down?

f'' < 0 or f' is decreasing

What is a point of inflection?

f'' changes sign (concavity changes).

What is the definition of the derivative?

ext{lim (h→0) } \frac{f(x+h) - f(x)}{h}

What is the alternate form of the definition of the derivative at x=c?

ext{lim (x→c) } \frac{f(x) - f(c)}{x - c}

What is the equation of the tangent line at x = a?

y = f(a) + f'(a)(x - a)­

What does L'Hôpital's Rule state?

If ext{lim (x→a) } \frac{f(x)}{g(x)} is indeterminate, then ext{lim (x→a) } \frac{f(x)}{g(x)} = \text{lim (x→a) } \frac{f'(x)}{g'(x)}

What is the product rule?

f'(x)g(x) + f(x)g'(x)

What is the quotient rule?

\frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

What is the derivative of an inverse function?

(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))} or g'(x) = \frac{1}{f'(g(x))}

What does the Mean Value Theorem state?

If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists a number c in (a, b) such that f'(c) = \frac{f(b) - f(a)}{b - a}

Velocity =

s'(t)

Acceleration =

= v'(t)=s''(t)

Speed (parametric) =

\text{√}((x'(t))^2 + (y'(t))^2)

Polar Coordinates

x = r cos θ, y = r sin θ

Pythagorean Identities

cos²x + sin²x = 1

1+tan²x = sec²x

1 + cot²x = csc²x

Alternating Remainder

|Rn| = |S - Sn

Chain Rule

If y = f(g(x)), then y' = f'(g(x))g'(x)

OR

\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} y}{\mathrm{d} u} \cdot \frac{\mathrm{d} u}{\mathrm{d} x}

Derivative of xⁿ

nx^n-1

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

tan(x)

sec²(x)

cot(x)

-csc²(x)

sec(x)

sec(x)tan(x)

csc(x)

-csc(x)cot(x)

arcsin(u)

\frac{u^{\prime}}{\sqrt{1-u^2}}

arccos(u)

\frac{-u^2}{\sqrt{1-u^2}}

arctan(u)

\frac{u^{\prime}}{1+u^2}

arccot(u)

\frac{-u^{\prime}}{1+u^2}

arcsec(u)

\frac{u^{\prime}}{\left|u\right|\sqrt{u^2-1}}

arccsc(u)

\frac{-u^2}{\left|u\right|\sqrt{u^2}-1}

Derivative of e^{u}

e^{u} du

a^{u} e of ln(u)

\frac{u^{\prime}}{u}

Derivative of a^{u}

a^{u} ln(a)u’

Derivative of \log_{a}u

\frac{u^{\prime}}{u\ln a}

Rolle’s Theorem

If f(x) is continuous on [a,b] and differentiable on (a,b) and f(a) = f(b), then there exists a number “c” on (a,b) such that f’(c ) = 0.

The Fundamental Theorem of Calculus

\int_{a}^{b}f\left(x\right)\!\,dx=F\left(b\right)-F\left(a\right)

2nd FTC

\frac{d}{dx}\int_{a}^{x}f(t)dt=f(x)

\frac{d}{dx}\int_{a}^{g(x)}f(t)dt=f(g(x))g'(x)

Mean Value Theorem for Integrals

f(c)=\frac{1}{b-a}\int_{a}^{b}f(x)dx

Solids of Revolution and Friends:

Disk Method

v=\pi\int_{a}^{b}\left[ R\left(x\right)\right]^2dx

OR

v=\pi\int_{c}^{d}\left[ R(y)^2\right]dy

Solids of Revolution and Friends:

Washer Method

v=\pi \int_{a}^{b} \left(R(x)^2 - r(x)^2\right) dx or dy

Solids of Revolution and Friends:

General Volume Equation

v=\int_{a}^{b} \left(Area\right) dx

Solids of Revolution and Friends:

General Volume Equation

v=\int_{a}^{b} \left(Area\right) dx

Arc Length (Rectangular)

\int_{a}^{b}\sqrt{1+[f'(x)]^2} dx or g’(y) and dy.

Arc Length (Parametric)

\int{t_1}^{t_2} \sqrt{ (x'(t))^2 + (y'(t))^2 } dt

Arc Length (Polar)

\int_{a}^{b}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta

Distance, Velocity, and Acceleration
S9T) is the ___ function?

Position

Distance, Velocity, and Acceleration

<x(t),y(t)> is the ___ in parametric?

Position

Distance, Velocity, and Acceleration

Velocity = ___

S’(t)

Distance, Velocity, and Acceleration

Acceleration = ___

v’(t) = s’’(t)

Distance, Velocity, and Acceleration

Velocity Vector = ___

<x’(t), y’(t)>

Distance, Velocity, and Acceleration

Acceleration Vector = ___

<x’’(t), y’’(t)>

Distance, Velocity, and Acceleration

speed (parametric)

|v(t)| = ___

\sqrt{\left\lbrack x^{\prime}\left(t\right)\left\rbrack^2+\right.\left\lbrack y^{\prime}\left(t\right)\right\rbrack^2\right.}

Distance, Velocity, and Acceleration

Displacement = ___

\int_{a}^{b}v(t)dt

Distance, Velocity, and Acceleration

Distance (rectangular and parametric):

\int_{a}^{b}\!\left|v\left(t\right)\right|\,dt=\int_{a}^{b}\!\left(speed\right)\,dt

Distance, Velocity, and Acceleration

Average Velocity = ___

\frac{S\left(b\right)-S\left(a\right)}{b-a}

and

\frac{1}{b-a}\int_{a}^{b}\!v\left(t\right)\,dt

Euler’s Method

n	x_{n}	y_{n}=y_{n-1}+hF\left(x_0,y_0\right)
0	0	given
.	.	x_{n}=x_{n-1}+h,h=\Delta x

Integration by Parts

\int_{}^{}\!u\,dv=uv-\int_{}^{}\!v\,du

Logistics

dy/dt = ___

ky\left(1-\frac{y}{L}\right)

Logistics

y = ___

\frac{L}{1+Ce^{-kt}}

Logistics

C = ___

\frac{L-y_0}{y_0}

Logistics

\lim_{t\rightarrow\infty}y\left(t\right)= ___

L

Logistics

What does y represent?

Population

Logistics

When is population increasing fastest?

\frac{L}{2}

Parametric Equations

\frac{dy}{dx} = ?

\frac{\frac{dy}{dt}}{\frac{dx}{dt}}

Parametric Equations

\frac{d^2y}{dx^2} = ?

\frac{\frac{d}{dt}\left[\frac{dy}{dx}\right]}{\frac{dx}{dt}}

Polar Curves

Area = ?

\int_{a}^{b}\frac12r^2\,d\theta

Polar Curves

Slope = \frac{\mathbb{d}y}{\mathbb{d}x} = ?

\frac{\mathbb{d}y}{\mathbb{d}x} = \frac{\frac{\mathbb{d}y}{\mathbb{d}\theta}}{\frac{\mathbb{d}x}{\mathbb{d}\theta}} = \frac{\frac{\mathbb{d}r}{\mathbb{d}\theta} \sin(\theta) + r \cos(\theta)}{\frac{\mathbb{d}r}{\mathbb{d}\theta} \cos(\theta) - r \sin(\theta)}

Taylor Series

f(c) + f'(c)(x-c) + \frac{f''(c)(x-c)^2}{2!} + \dots = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)(x-c)^n}{n!}

Maclaurin Series

f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \cdots

Maclaurin Series

e^{x}=?

1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^{n}}{n!}

Maclaurin Series

\cos x = ?

1-\frac{x^2}{2!}+\frac{x^4}{4!}-...+\frac{\left(-1\right)^{n}x^2n}{\left(2n\right)!}

Maclaurin Series

\sin x = ?

x-\frac{x^3}{3!}+\frac{x^5}{5!}-...+\frac{\left(-1\right)^{n}x^{2n-1}}{\left(2n+1\right)!}

Maclaurin Series

\frac{1}{1-x} = ?

1+x+x^2+x^3+...+x^{n}

Maclaurin Series

\ln\left(x+1\right) = ?

x-\frac{x^2}{2}+\frac{x^3}{3}-...+\frac{\left(-1\right)^{n-1}x^{n}}{n}

Alternating Remainder

\left|R_{n}\right|=?

If the alternating series converges to 0 anda_{n+1}\le a_{n} , then

\left|S-S_{n}\right|\le a_{n+1}\larr 1st neglected term

Lagrange Error

\left|R_{n}\left(x\right)\left|\le\right.\right. ?

max value off^{\left(n+1\right)}on\left\lbrack x,c\right\rbrack\left(x-c\right)^{n+1}

---

\left(n+1\right)!

Trig Identities: Double Angles

sin(2x) = ?

2\sin x\cos x

Trig Identities: Double Angles

cos(2x) = ?

\cos^2\left(x\right)-\sin^2\left(x\right)

2\cos^2\left(x\right)-1

1-\sin^2\left(x\right)

Trig Identities: Double Angles

\sin^2\left(x\right)=?

\frac{1-\cos\left(2x\right)}{2}

Trig Identities: Double Angles

\cos^2\left(x\right)=?

\frac{1+\cos\left(2x\right)}{2}

Trig Identities: Reciprocal Identities

csc(x) = ?

\frac{1}{\sin\left(x\right)}

Trig Identities: Reciprocal Identities

sec(x) = ?

\frac{1}{\cos\left(x\right)}

Trig Identities: Reciprocal Identities

cot(x)

\frac{1}{\tan\left(x\right)}

Trig Identities: Ratio Identities

tan(x) = ?

\frac{\sin\left(x\right)}{\cos\left(x\right)}

Trig Identities: Ratio Identities

cot(x) = ?

\frac{\cos\left(x\right)}{\sin\left(x\right)}

Trig Identities: Pythagorean Identities

\sin^2\left(x\right)+\cos^2\left(x\right)=?

1

Trig Identities: Pythagorean Identities

1+\tan^2\left(x\right)=?

\sec^2\left(x\right)

Trig Identities: Pythagorean Identities

1+\cot^2\left(x\right)

\csc^2\left(x\right)

Other Integration Rules:

\int_{}^{}\!x^{n}\,dx=?

\frac{1}{n+1}x^{n+1}+c

Other Integration Rules:

\int_{}^{}\!\tan u\,du=?

\ln\left|\sec\left(u\right)\right|+c or -\ln\left|\cos\left(u\right)\right|+c

Other Integration Rules:

\int_{}^{}\!\sec\left(u\right)\,du

\ln\left|\sec\left(u\right)+\tan\left(u\right)\right|+c