Barron's AP Calc AB+BC Flashcards (Units 1 - 5)

Deck Includes: - Key Formulae - Basic Derivatives + Rules for taking Derivatives - Basic Integrals + Rules for taking Anti-Derivatives (more will be added as I progress through the book)

Unit 1: Functions

Precalc Review of Properties of:

• Rational Functions

• Trig/Inverse Trig Functions

• Exponential/Logarithmic Functions

• Parametric Functions (BC)

• Polar Curves (BC)

Even Function

f(-x) = -f(x)

Odd Function

f(-x) = f(x)

Rational Function

f(x) = \frac{p(x)}{q(x)}
(Domain: set of all real # where q(x) \neq 0 )
(Range: (-\infty, +\infty) )

State the Domain and Range for all Trig Functions

State the Domain and Range for all Inverse Trig Functions

Exponential Functions

f(x) = a^x (a>0, a\neq 1)

Logarithmic Function

y=\log_ax IF AND ONLY IF a^y=x

Write down all the log properties you remember

• \log_a1=0

• \log_aa=1

• log_a a^m = m

• \log_amn=\log_am+\log_an (Product Rule)

• \log_a\frac{m}{n} = \log_am-\log_an (Quotient Rule)

• \log_ax^m=m\log_ax (Exponent Rule)

• \log_mn=\frac{\log_an}{\log_am} (Change of Base Rule)

Parametric Equation

x=f(t), y=g(t) where t is the parameter

Polar Function

r=f(\theta)
(very broad def.)

Write down all the types of polar functions you remember (BC)

• Spiral: r=\theta

• Rose Curve: r= k \cdot sin(m \cdot \theta) or r = k \cdot cos(m \cdot \theta)

• Limacons: r= k + m \cdot sin(\theta) or r= k + m \cdot cos (\theta) -

  • Dimpled Limacons: k>m

  • Cardioid: k=m

  • Limacon w/Inner Loop: k<m

Unit 2: Limits and Continuity

• general properties of limits

• how to find limits

• horizontal and vertical asymptotes

• continuity

• types of discontinuities (removable, jump, infinite)

• Squeeze Theorem

• Extreme Value Theorem

• Intermediate Value Theorem

Limit and One-Sided Limit

• \lim_{x \to c} f(x) = L

  (as x approaches c, f(x) = L )

• \lim_{x \to c^+} f(x) = L or \lim_{x \to c^-} f(x) = L

  (as x approaches c from the left or right, f(x)=L )

Horizontal Asymptote and Vertical Asymptote

• \lim_{x \to \pm \infty}f(x)=b

  (horizontal asymptote)

• \lim_{x \to a}f(x)=\pm \infty OR \lim_{x \to a^ \pm} f(x) = \pm \infty

  (vertical asymptote)

Write down all the properties of limits you remember

if \lim f(x) and \lim g(x) are finite numbers (not going to \pm \infty ), then

• \lim k \cdot f(x) = k \cdot \lim f(x)

• \lim [f(x) + g(x)] = \lim f(x) \cdot \lim g(x)

• \lim (f(x) \cdot g(x)) = \lim f(x) \cdot \lim g(x)

• \lim {\frac {f(x)}{g(x)}} = \frac{\lim f(x)}{\lim g(x)}

• \lim_{x \to k} = k

Sandwich (Squeeze) Theorem

if f(x) \ge g(x) \ge h(x) AND
if \lim_{x \to c} = \lim_{x \to c}=L,
then lim_{x \to c} = L

Rational Function Theorem

• if the degree of p(x) < q(x), then \lim_{x \to \pm \infty} \frac{p(x)}{q(x)}= 0
  (horizontal asymptote)

• if the degree of p(x) = q(x), then \lim_{x \to \pm \infty} \frac{p(x)}{q(x)}= \frac{a_p}{b_q} , where a and b are the coefficients of the highest powers
  (horizontal asymptote)

• if the degree of p(x) > q(x), then \lim_{x \to \pm \infty} \frac{p(x)}{q(x)}= \pm \infty (vertical asymptote/DNE)

What is the limit of sin \theta?

\lim_{x \to \theta} \frac {sin \theta}{\theta} = 1
(if \theta is in rads)

Continuous

• f(a) exists

• \lim_{x \to a} f(x) exists

• \lim_{x \to a} f(x) = f(a)

Define the three types of discontinuities

• Removable Discontinuity: \lim_{x \to a} f(x) AND f(a) exists, but \lim_{x \to a} f(x) \ne f(a)

• Jump Discontinuity: \lim_{x \to a^+} f(x) \ne \lim_{x \to a^-} f(x)

• Infinite Discontinuity: \lim_{x \to a^-} f(x) = \pm \infty

Extreme Value Theorem

if f is continuous in [a, b], then f has a min and max value in that interval

Intermediate Value Theorem

if f is continuous in [a, b], and f(a) \le m \le f(b), then there is at least one number d in [a, b] that f(d) = m

• if f(a) and f(b) have opposite signs then there is a value d in [a, b] where f(d) = 0.

Continuous Function Theorem

if f(x) and g(x) are both continuous at x=c, the following will also be continuous:

• k \cdot f(x)

• f(x) \pm g(x)

• f(x) \cdot g(x)

• \frac{f(x)}{g(x)}, if g(x) \neq 0

Unit 3: Differentiation

• definitions of derivatives

• estimating derivatives

• derivatives of basic functions

• product, quotient, chain rules

• implicit differentiation

• Rolle’s Theorem

• Mean Value Theorem

• L’Hopital’s Rule

• derivatives of parametric functions (BC)

Derivative

• \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}, often notated as f’(x), y’, \frac{dy}{dx}, or D_xy

Difference Quotient

• \frac{f(a+h)-f(a)}{h}

• Represents the avg. rate of change from a to a+h.

Symmetric Difference Quotient

f’(a) \simeq \frac{f(a+h)-f(a-h)}{2h}

Derivatives of Constants and Coefficients

(a and n are constants and u and v are differentiable functions of x)

• \frac{da}{dx}=0

• \frac{d}{dx}au=a\frac{du}{dx}

Chain Rule

\frac{dy}{du} \cdot \frac{du}{dx}

“Take the derivative of the “outside” function w/the “inside” function left the same, then multiply by the derivative of the “inside” one”

^^very important concept, make sure you understand!

Power Rule

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx}u^a=au^{a-1}\frac{du}{dx}

Addition and Subtraction Rules

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx}(u+v)=\frac{d}{dx}u+\frac{d}{dx}v

• \frac{d}{dx}(u-v)=\frac{d}{dx}u-\frac{d}{dx}v

Product Rule

(a and n are constants and u and v are differentiable functions of x)

\frac{d}{dx}(u \cdot v)=u \frac{dv}{dx}+v \frac{du}{dx}

Quotient Rule

\frac{d}{dx}(\frac{u}{v})=\frac{{v \frac{du}{dx}}-u \frac{du}{dx}}{v²}, (v \neq 0)

Sine and Cosine Derivatives

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx} \sin u = \cos u \frac{d}{dx}

• \frac{d}{dx} \cos u = -\sin u \frac{d}{dx}

Tangent and Cotangent Derivatives

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx} \tan u = \sec²u \frac{du}{dx}

• \frac{d}{dx} \cot u = -\csc²u \frac{du}{dx}

Secant and Cosecant Derivatives

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx} \sec u = \sec u \tan u \frac{du}{dx}

• \frac{d}{dx} \csc u = -\csc u \cot u \frac{du}{dx}

Exponent and Natural Log Derivatives

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx}a^u = a^u \ln a \frac{du}{dx}

• \frac{d}{dx} e^u = e^u \frac{du}{dx}

• \frac{d}{dx} \ln u = \frac{1}{u} \cdot \frac{du}{dx}

Arcsin and Arccos Derivatives

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx} \sin^{-1}u = \frac{1}{\sqrt{1-u²}} \cdot \frac{du}{dx}, (-1<u<1)

• \frac{d}{dx} \cos^{-1}u= -\frac{1}{\sqrt{1-u²}}, (-1<u<1)

Arctan and Arccot Derivatives

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx} \tan^{-1}u = \frac{1}{1+u²} \cdot \frac{du}{dx}

• \frac{d}{dx} \cot^{-1}u = -\frac{1}{1+u²} \cdot \frac{du}{dx}

Arcsec and Arccsc Derivatives

(a and n are constants and u and v are differentiable functions of x)

• \frac{d}{dx} \sec^{-1}u = \frac{1}{|u| \sqrt{u²-1}} \cdot \frac{du}{dx}, (|u|>1)

• \frac{d}{dx} \csc^{-1}u = -\frac{1}{|u| \sqrt{u²-1}} \cdot \frac{du}{dx}, (|u|>1)

What is the relationship between differentiability and continuity?

“Differentiability implies continuity, but continuity does not imply differentiability”

Parametric First Derivative (BC)v

\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} where t is the parameter

Parametric Second Derivative (BC)

\frac{d²y}{dx²} = \frac{d}{dx}(\frac{dy}{dx}) = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}

Unit 4: Applications of Differential Calculus

• slopes of curves/eqs. of tangent lines

• maxima, minima, points of inflections

• increasing, decreasing, concave up and down

• motion along a line

• local linear approximations

• related rates

• slope of parametric and polar curves (BC)

• motion along parametrically defined curves (BC)

Slope of a curve

slope of the tanget to the curve at point (x_1, y_1) is f’(x) at x=x_1

Critical Point

• f’(a)=0 or f’(a)= undefined

• if f has a derivative everywhere, solve f’(x)=0

Equation of the tangent to a curve

(curve =y=f(x) at point P(x_1, y_1)) is:

y-y_1=f’(x_1)(x-x_1)

• if the tangent is horizontal at a point, then the derivative = 0 (no slope=no rate of change=no derivative)

• if the tangent is vertical at a point, then the derivative does not exist

Tangents to Parametrically Defined Curves