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Cards (123)

Section 1

(50 cards)

ln 1

Front

0

Back

Volume of a Sphere

Front

V = 4/3 pi r^3

Back

ln (mn)

Front

ln m + ln n

Back

cos π/4

Front

√2/2

Back

cos π/2

Front

0

Back

Volume of a cone

Front

πr²h/3

Back

1 + cot²x

Front

csc²x

Back

sin π/4

Front

√2/2

Back

Continuity & differentiability

Front

Differentiability implies continuity, but continuity does not necessarily imply differentiability.

Back

Surface Area of a Sphere

Front

S = 4 pi r^2

Back

Circumference of a circle

Front

2πr

Back

sin π/6

Front

1/2

Back

sin(2x)

Front

2 sin x cos x

Back

sin π/3

Front

√3/2

Back

cos π/6

Front

√3/2

Back

sin 0

Front

0

Back

cos 3π/2

Front

0

Back

1 + tan²x

Front

sec²x

Back

Continuity on a closed interval, [a,b]

Front

1. f(x) is continuous on the closed interval (a,b) 2. The limit from the right as x approaches a of f(x) is f(a) 3. The limit from the left as x approaches b of f(x) is f(b)

Back

The limit as x approaches 0 of sin x / x

Front

1

Back

sin π/2

Front

1

Back

Limit Definition of a Derivative

Front

f'(x) = lim as ∆x → 0 of [ f(x + ∆x) - f(x) ] / ∆x

Back

Continuity at a point (x = c)

Front

1. f(x) is defined at f(c) 2. The limit as x approaches c of f(x) exists 3. The limit as x approaches c of f(x) = f(c)

Back

If f(-x) = f(x)

Front

f is an even function

Back

ln (m/n)

Front

ln m - ln n

Back

Area of an equilateral triangle

Front

√3s² / 4

Back

If f(-x) = -f(x)

Front

f is an odd function

Back

Continuity on an open interval, (a,b)

Front

f(x) is continuous if for every point on the interval (a,b) the conditions for continuity at a point are satisfied.

Back

The limit as x approaches 0 of (1 - cos x) / x

Front

0

Back

csc x

Front

1 / sin x

Back

cos(2x)

Front

cos²x - sin²x

Back

cos π/3

Front

1/2

Back

ln e

Front

1

Back

tan x

Front

sin x / cos x

Back

sin²x

Front

(1 - cos 2x) / 2

Back

sec x

Front

1 / cos x

Back

sin 3π/2

Front

-1

Back

How to get from precalculus to calculus

Front

Limits

Back

Alternate Limit Definition of a derivative

Front

f'(x) = lim as x → c of [ f(x) - f(c) ] / [ x - c]

Back

Area of a circle

Front

πr²

Back

Volume of a right circular cylinder

Front

πr²h

Back

cos π

Front

-1

Back

Indeterminate form

Front

0/0

Back

cos 0

Front

1

Back

cot x

Front

1 / tan x = cos x / sin x

Back

cos²x + sin²x

Front

1

Back

cos²x

Front

(1 + cos 2x) / 2

Back

Intermediate Value Theorem

Front

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

Back

sin π

Front

0

Back

ln mⁿ

Front

n ln m

Back

Section 2

(50 cards)

d/dx[log_a u]

Front

u'/((ln a) u)

Back

d/dx[cot x]

Front

-csc² x

Back

Instantaneous velocity

Front

Derivative of position at a point

Back

∫k f(x) dx

Front

k ∫ f(x) dx

Back

d/dx[arccsc x]

Front

-1/(|x|√(x²-1))

Back

d/dx[sec x]

Front

sec x tan x

Back

Mean Value Theorem

Front

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)

Back

d/dx[arccot x]

Front

-1/(1+x²)

Back

d/dx[x]

Front

1

Back

d/dx[arcsin x]

Front

1/√(1-x²)

Back

d/dx[csc x]

Front

-csc x cot x

Back

The Product Rule

Front

If two functions, f and g, are differentiable, then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

Back

Guidelines for implicit differentiation

Front

1. Differentiate both sides w.r.t. x 2. Move all y' terms to one side & other terms to the other 3. Factor out y' 4. Divide to solve for y'

Back

Derivative of an inverse (if g(x) is the inverse of f(x))

Front

g'(x) = 1/f'(g(x)), f'(g(x)) cannot = 0

Back

d/dx[arctan x]

Front

1/(1+x²)

Back

d/dx[log_a x]

Front

1/((ln a) x)

Back

∫cos x dx

Front

sin x + C

Back

Position function of a falling object (with acceleration in m/s²)

Front

s(t) = -4.9t²+ v₀t + s₀, v₀ = initial velocity, s₀ = initial height

Back

d/dx[ln u]

Front

u'/u, u > 0

Back

d/dx[ f(x) g(x) ]

Front

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

Back

Average speed

Front

∆s/∆t

Back

d/dx[sin x]

Front

cos x

Back

d/dx[e^u]

Front

u' e^u

Back

∫sin x dx

Front

-cos x + C

Back

Derivative of a constant

Front

d/dx[c] = 0

Back

Derivative

Front

Slope of a function at a point/slope of the tangent line to a function at a point

Back

d/dx[arcsec x]

Front

1/(|x|√(x²-1))

Back

Rolle's Theorem

Front

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

Back

Extreme Value Theorem

Front

If f is continuous on the closed interval [a,b] then it must have both a minimum and maximum on [a,b].

Back

Sum and Difference Rules for Derivatives

Front

d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

Back

d/dx[cos x]

Front

-sin x

Back

∫0 dx

Front

C

Back

FUNdamental Theorem of Calculus

Front

∫ f(x) dx on interval a to b = F(b) - F(a)

Back

d/dx[ln x]

Front

1/x, x>0

Back

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

Front

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

Back

Constant Multiple Rule for Derivatives

Front

d/dx[cf(x)] = c f'(x)

Back

d/dx[ f(x) / g(x) ]

Front

[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

Back

Position function of a falling object (with acceleration in ft/s²)

Front

s(t) = -16t²+ v₀t + s₀, v₀ = initial velocity, s₀ = initial height

Back

∫k dx

Front

kx + C

Back

d/dx[tan x]

Front

sec² x

Back

Velocity, v(t)

Front

Derivative of Position, s'(t)

Back

d/dx[a^u]

Front

u' (ln a) a^u

Back

∫xⁿ dx

Front

xⁿ⁺¹ / (n + 1) + C

Back

Power Rule for Derivatives

Front

d/dx[x^n]=nx^(n-1)

Back

d/dx[arccos x]

Front

-1/√(1-x²)

Back

Critical number

Front

x values where f'(x) is zero or undefined.

Back

d/dx[a^x]

Front

(ln a) a^x

Back

d/dx[e^x]

Front

e^x

Back

The Quotient Rule

Front

If two functions, f and g, are differentiable, then d/dx[ f(x) / g(x) ] = [g(x)f'(x) - f(x) g'(x)] / [g(x)]²

Back

Guidelines for solving related rates problems

Front

1. Given, Want, Sketch 2. Write an equation using variables given/to be determined 3. Differentiate w.r.t. time (using chain rule) 4. Plug in & solve

Back

Section 3

(23 cards)

∑ from i = 1 to n of i²

Front

n(n+1)(2n + 1)/6

Back

∑ from i = 1 to n of c

Front

cn

Back

∫e^x dx

Front

e^x+C

Back

∫ a^x dx

Front

a^x/ ln a + C

Back

∫ csc udu

Front

-ln |cscu + cotu| + C

Back

∫ du / (a² + u²)

Front

1/a arctan (u/a) + C

Back

∑ from i = 1 to n of i³

Front

n²(n+1)²/4

Back

∑ from i = 1 to n of i

Front

n(n+1)/2

Back

∫ sec udu

Front

ln |sec u + tan u| +C

Back

∫(1/x) dx

Front

ln | x | + c

Back

∫ cot udu

Front

ln |sinu| + C

Back

Area between two curves in an interval [a,b] where f(x) > g(x) in the entire interval

Front

∫ [f(x) - g(x)] dx from a to b

Back

∫e^u du

Front

e^u+C

Back

Mean Value Theorem for Integrals; If f(x) is continuous, then there exists a c such that

Front

∫f(x) dx (from a to b) = f(c) (b-a)

Back

∫csc²x dx

Front

- cot x + C

Back

∫sec x tan x dx

Front

sec x + C

Back

∫ u'/u dx

Front

ln |u| + C

Back

Average Value of a Function

Front

1 / (b - a) ∫ f(x) dx (from a to b)

Back

∫csc x cot x dx

Front

- csc x + C

Back

∫ tan udu

Front

-ln |cosu| +C

Back

∫ du / √(a² - u²)

Front

arcsin (u/a) + C

Back

∫sec²x dx

Front

tan x + C

Back

∫ du / (u√(u² - a²))

Front

1/a arcsec (|u|/a) + C

Back