Calc stuff 2

$\frac{x^{n+1}}{n+1}$ + C

∫xⁿ dx =

$$\int\frac{1}{x}d\left(x\right)$$

ln|x| + C

∫ e^kx dx

$$\frac{e^{kx}}{k}+C$$

Chain Rule: d/dx [f(g(x))] =

f’(g) g’(x)

∫cos(kx) dx =

$\frac{1}{k}$ sin(kx) + c

∫sin(kx) dx =

− $\frac{1}{k}$ cos(kx) + c

∫sec² x dx =

tan(x) + c

∫csc(x) cot(x) dx =

-csc(x) + c

∫sec(x) tan(x) dx =

sec(x) + C

∫csc² x dx =

-cot(x) + C

Slope of the Secant Line

A measure of the slope of the tangent line, or the rate of change, of $f(x)$ at the given point $(a, f(a))$ calculated as $\frac{f(x) - f(a)}{x - a}$ .

Instantaneous Velocity

The limiting values of the average velocities over shorter and shorter time periods.

Limit of a Function

To say that $\lim_{x \to a} f(x) = L$ means that as $x$ approaches $a$, but $x \neq a$, then $f(x)$ must approach $L$.

One-Sided Limits

A limit where the value is different when approaching from either the positive side ($x \to a^+$) or the negative side ($x \to a^-$).

Infinite Limit Theorem (Positive Even Integer)

If $n$ is a positive even integer, then $\lim_{x \to a^+} \frac{1}{(x-a)^n} = \infty$, $\lim_{x \to a^-} \frac{1}{(x-a)^n} = \infty$, and $\lim_{x \to a} \frac{1}{(x-a)^n} = \infty$.

Infinite Limit Theorem (Positive Odd Integer)

If $n$ is a positive odd integer, then $\lim_{x \to a^+} \frac{1}{(x-a)^n} = \infty$ and $\lim_{x \to a^-} \frac{1}{(x-a)^n} = -\infty$, hence the two-sided limit is DNE (Does Not Exist).

Sum Law Limits

The limit law stating $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$.

Limits Product Law

The limit law stating $\lim_{x \to a} [f(x) g(x)] = (\lim_{x \to a} f(x)) (\lim_{x \to a} g(x))$.

Direct Substitution Property

If $f$ is a function such that $a$ is in the domain of $f$, then $\lim_{x \to a} f(x) = f(a)$.

Squeeze Theorem

If $f(x) \leq g(x) \leq h(x)$ when $x$ is near $a$ (but not necessarily equal to $a$) and $\lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)$, then $\lim_{x \to a} g(x) = L$.

Continuity at a Point

A function $f$ is continuous at $a$ if:
1. $f(a)$ is defined;
2. $\lim_{x \to a} f(x)$ exists;
3. $\lim_{x \to a} f(x) = f(a)$.

Limit Definition of Derivative

The derivative of a function at a number $a$, denoted by $f'(a)$, is defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

Differentiable at a Number

A function $f$ is differentiable at a number $a$ if the limit $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists.

Differentiability Implies Continuity

If $f(x)$ is differentiable at $a$, then $f$ is continuous at $a$.

The Power Rule Derivatives

The differentiation rule stating that for any real number $n$, $\frac{d}{dx} [x^n] = n x^{n-1}$.

The Product Rule

The differentiation rule stating that $\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$.

The Quotient Rule

The differentiation rule stating that $\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}$.

Particle at Rest

The state of a moving particle when its velocity at time $t$ is equal to zero ($v(t) = 0$).

Horizontal Tangent Line

A line on a function $f(x)$ where the slope is zero, found by setting the derivative $f'(x) = 0$.

Big Three Limits

1. $\lim_{x\to0}\frac{\sin(ax)}{ax}=1$ ;

2. $\lim_{x\to0}\frac{1-\cos\left(ax\right)}{ax}=0$ ;

3. $\lim_{x \to 0} \frac{\tan(x)}{x} = 1$ .

$$\frac{d}{d\left(x\right)}\ln\left(g\left(x\right)\right)$$

$$\frac{g^{\prime}\left(x\right)}{g\left(x\right)}$$

$$\frac{d}{d\left(x\right)}\left(\log_{a}\left(x\right)\right)$$

$\frac{1}{x\ln\left(a\right)},x>0$ .

$$\frac{d}{d\left(x\right)}\left(\log_{a}g\left(x\right)\right)$$

$$\frac{g^{\prime}\left(x\right)}{g\left(x\right)\ln a}$$

$$\frac{d}{d\left(x\right)}\left(e^{x}\right)$$

$$e^{x}$$

$$\frac{d}{\left(dx\right)}\left(a^{x}\right)$$

$$a^{x}\ln\left(a\right)$$

$$\frac{d}{d\left(x\right)}\left(e^{g\left(x\right)}\right)$$

$$e^{g\left(x\right)}g^{\prime}\left(x\right)$$

$$\frac{d}{d\left(x\right)}\left(a^{g\left(x\right)}\right)$$

$$\ln\left(a\right)a^{g\left(x\right)}g^{\prime}\left(x\right)$$

$\frac{d}{d\left(x\right)}\cos^{-1}\left(x\right)$ =

$$\frac{-1}{\sqrt{1-x^2}}$$

$\frac{d}{d\left(x\right)}\sin^{-1}\left(x\right)$ =

$$\frac{1}{\sqrt{1-x^2}}$$

$\frac{d}{d\left(x_{}\right)}\cot^{-1}\left(x\right)$ =

$$\frac{-1}{1+x^2}$$

$\frac{d}{d\left(x\right)}\csc^{-1}\left(x\right)$ =

$$\frac{-1}{|x|\sqrt{1-x^2}}$$

$\frac{d}{d\left(x\right)}\tan^{-1}\left(x\right)$ =

$$\frac{1}{1+x^2}$$

$$\frac{d}{d\left(x\right)}\sec^{-1}\left(x\right)$$

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

sec²(x) =

1 + tan²(x)

d/dx (1/x) =

$$-\frac{1}{x^2}$$

$$\frac{d}{d\left(x\right)}\left(\sqrt{x}\right)=$$

$$\frac{1}{2\sqrt{x}}$$

Particle is slowing down:

v(t) and a(t) have different signs

Particle is moving left/down:

v(t) < 0 (negative)

Particle is moving right/up:

v(t) > 0 (positive)

Particle is speeding up: (|velocity| is getting bigger)

v(t) and a(t) have same sign

Intermediate Value Theorem

If f is continuous on [a,b] and k is any number between f(a) and f(b), then there is at least one number c between a and b such that f(c) = k

Steps for Absolute Mins/Max:

1. Find critical numbers using derivative

2. Identify endpoints

3. f(critical point) and f(end point)

4. Determine absolute max/min values by comparing the y-values, stated in a sentence

Mean Value Theorem

$$f^{\prime}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

If f is continuous on [a,b] and differentiable on (a,b), then there exists a number “c” on (a,b):

x = c if a critical number because

f’(x) = 0 or f’(x) is undefined

f(x) is ____ on [a,b] because f’(x) > 0

increasing

f(x) is ____ on [a,b] because f’(x) < 0

decreasing

f(x) is ____ on (a,b) because f’’(x) > 0

concave up

f(x) is ____ on (a,b) because f’’(x) < 0

concave down

f’’(x) = 0 or = DNE and f“(x) changes sign around point c. (Or if f’(x) slope changes sign and f’’(x) = 0 or DNE)

Inflection point

$$∫\tan xdx=$$

$$-\ln\left|\cos x\right|+C$$

$$\int\left(\frac{1}{\left(x^2+a^2\right)}\right)d\left(x\right)$$

$$\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C$$

F.T.O.C :$\int_{a}^{b}\!f^{\prime}\left(x\right)\,dx=$

$$f\left(b\right)-f\left(a\right)$$

$$\int_{a}^{b}\!f\left(x\right)+g\left(x\right)\,dx=$$

$$\int_{a}^{b}\!f\left(x\right)d\left(x\right)+\int_{a}^{b}\!g\left(x\right)d\left(x\right)$$

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

$$\int_{a}^{b}\!f\left(x\right)\,d\left(x\right)-\int_{a}^{b}\!g\left(x\right)d\left(x\right)$$

$$\int_{a}^{b}\!cf\left(x\right)\,dx=$$

$$c\int_{a}^{b}\!f\left(x\right)\,dx$$

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

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

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

0

Average Value of a Function

$$\frac{1}{b-a}\int_{a}^{b}\!f\left(x\right)d\left(x\right)$$

Second F.T.O.C. P1:

$\frac{d}{d\left(x\right)}\int_{a}^{x}\!f\left(t\right)\,d\left(t\right)=$ f(x)

Second F.T.O.C. P2:

$\frac{d}{d\left(x\right)}\int_{a}^{g\left(x\right)}\!f\left(t\right)d\left(t\right)=$ $\!f\left(g\left(x\right)\right)\cdot\,g^{\prime}\left(x\right)$

$\int_{a}^{b}\!f\left(x\right)\,dx$ represents the ___ ______ in the function f from time a to b.

net change

Steps to solve differential equations.

1. separate variables

2. integrate each side

3. make sure to place C on side with independent variable (usually x)

4. plug initial condition and solve for C (if given)

5. solve for dependent variable (usually y)

$y=Ce^{kt}$ or $y=y_0e^{kt}$

Exponential growth

y= end amount

C or y0 = initial amount

k = growth constant/growth rate

t = time elapsed

$$y=y_0e^{-kt}$$

Exponential decay

negative sign = decays over time, rather than growing.

Velocity

v(t) = s’(t) = ∫a(t)dt

acceleration

a(t) = v’(t) = s’’(t)

speed

|v'(t)|

average velocity

$\frac{s\left(b\right)-s\left(a\right)}{b-a}$ (given v(t)); or $\frac{1}{b-a}\int_{a}^{b}\!v\left(t\right)\,dt$ (given a(t))

average acceleration

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

Displacement

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

total distance

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

position at b

$$s\left(b\right)=s\left(a\right)+\int_{a}^{b}\!v\left(t\right)\,dt$$

Area perpendicular to x-axis; f(x) is top curve, g(x) is bottom, a and b are x-coordinates of point of intersection.

$$\int_{a}^{b}\!\left\lbrack f\left(x\right)-g\left(x\right)\right\rbrack\,dx=$$

Area perpendicular to y-axis; f(y) is right curve, g(y) is left curve, a and b are y-coordinates of point of intersection.

$$\int_{a}^{b}\!\left\lbrack f\left(y\right)-g\left(y\right)]d\left(y\right)=\right.$$

Volume steps

1. Decide whether it’s a dx or dy

2. Find a formula in terms of x or y

3. Find the limits (make sure they match x or y)

4. integrate and evaluate

Volume = ∫Area

Volume of a disc around a Horizontal Axis of rotation

V = $\int_{a}^{b}\!\pi r^2dx$

Volume of a washer around a horizontal axis of rotation.

$$\int_{a}^{b}\!\left\lbrack\pi R^2-\pi r^2\right\rbrack d\left(x\right)$$

$$\int_{a}^{b}\!A\left(x\right)\,dx$$

Volume of a slab (cross-section) around a horizontal axis of rotation. A(x) is the area for the cross section.

Volume of a disc around a vertical axis of rotation

V = $\int_{a}^{b}\!\pi r^2dy$

Volume of a washer around a vertical axis of rotation

Volume of a slab (cross-section) around a vertical axis of rotation. A(y) is the area for the cross section.

$$\int_{a}^{b}\!A\left(y\right)\,dy$$

Volume using shells

V = ∫2πr h (Thickness)

For volume rotated around a vertical axis (y-axis), using shells method.

$$\int_{a}^{b}\!2\pi\left(r\right)\left(h\left(x\right)\right)d\left(x\right))$$

For volume rotated around a horizontal axis (x-axis), using shells method.

$$\int_{a}^{b}\!2\pi\left(r\right)\left(h\left(y\right)\right)d\left(y\right)$$