Calc Midterm - Term/Formulas (Memorization Sheet)

x⁰

1 (if x ≠ 0) Any number raised to 0 equals 1

x¹

x The power of 1 keeps the number the same

x⁻ⁿ

1/xⁿ (if x ≠ 0) Negative exponents move to the denominator

xᵐ·xⁿ

xᵐ⁺ⁿ Add exponents when multiplying with same base

(xᵐ)ⁿ

xᵐⁿ Multiply exponents when raising a power to a power

xᵐ ÷ xⁿ

xᵐ⁻ⁿ (if x ≠ 0) Subtract exponents when dividing with same base

(xy)ᵐ

xᵐyᵐ Distribute exponent across multiplication

(x/y)ⁿ

xⁿ/yⁿ (if y ≠ 0) Distribute exponent across division

xᵐ⁄ⁿ

ⁿ√(xᵐ) (if x ≥ 0, m ≥ 0, n > 0) Fractional exponents represent roots

logₐ(1)

0 Log of 1 is always 0

logₐ(a)

1 Log of the base equals 1

y

logₐ(x) ⇔ x

a^(logₐM)

M Exponent–log inverse property

logₐ(MN)

logₐM + logₐN Log of a product

logₐ(M/N)

logₐM − logₐN Log of a quotient

logₐ(Mˣ)

x·logₐM Power rule of logs

Change of base formula logₐM

logbM / logba

x² − y²

(x + y)(x − y) Difference of squares

x³ ± y³

(x ± y)(x² ∓ xy + y²) Sum/difference of cubes

sin(−x)

−sinx Sine is odd

cos(−x)

cosx Cosine is even

sin²θ + cos²θ

1 Pythagorean identity

tan²θ + 1

sec²θ Second Pythagorean identity

cot²θ + 1

csc²θ Third Pythagorean identity

sin²θ

(1 − cos(2θ))/2 Half-angle identity for sine

cos²θ

(1 + cos(2θ))/2 Half-angle identity for cosine

tan²θ

(1 − cos(2θ))/(1 + cos(2θ)) Half-angle identity for tangent

sin(2θ)

2sinθcosθ Double-angle formula for sine

cos(2θ)

cos²θ − sin²θ

tan(2θ)

2tanθ / (1 − tan²θ) Double-angle formula for tangent

sin(a + b)

sin a cos b + cos a sin b Sum formula for sine

cos(a + b)

cos a cos b − sin a sin b Sum formula for cosine

Angle 0

π/6 π/4 π/3 π/2

sinθ

0 1/2 √2/2 √3/2 1

cosθ

1 √3/2 √2/2 1/2 0

tanθ

0 √3/3 1 √3 undefined

cscθ

undef 2 √2 2√3/3 1

secθ

1 2√3/3 √2 2 undef

cotθ

undef √3 1 √3/3 0

arcsin(x)

Domain [−1, 1] → Range [−π/2, π/2]

arccos(x)

Domain [−1, 1] → Range [0, π]

arctan(x)

Domain (−∞, ∞) → Range (−π/2, π/2)

limₙ→∞ (1 + 1/n)ⁿ

e Definition of e

limₓ→0 (sinx / x)

1 Fundamental trig limit

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) and both f,h → L, then g → L

Intermediate Value Theorem

If f continuous on [a,b] and z between f(a),f(b), ∃c ∈ [a,b] s.t. f(c)

Constant rule: (c)'

0 Derivative of a constant is 0

Sum rule: (f + g)'

f' + g' Derivative of sum is sum of derivatives

Difference rule: (f − g)'

f' − g' Derivative of difference is difference of derivatives

Constant multiple: (cf)'

c·f' Pull constants outside

Product rule: (fg)'

f'g + fg' Product rule formula

Quotient rule: (f/g)'

(g f' − f g') / g² Derivative of a quotient

Chain rule: (f(g(x)))'

f'(g(x))·g'(x) Derivative of a composition

(xⁿ)'

n·xⁿ⁻¹ Power rule

(eˣ)'

eˣ Exponential rule

(aˣ)'

aˣ·ln(a) Exponential with base a

(ln|x|)'

1/x Logarithmic derivative

(logₐx)'

1/(x ln a) Derivative of log base a

(sin x)'

cos x Derivative of sine

(cos x)'

−sin x Derivative of cosine

(tan x)'

sec²x Derivative of tangent

(cot x)'

−csc²x Derivative of cotangent

(arcsin x)'

1/√(1 − x²) Derivative of arcsine

(arccos x)'

−1/√(1 − x²) Derivative of arccosine

(arctan x)'

1/(1 + x²) Derivative of arctangent

(arc cot x)'

−1/(1 + x²) Derivative of arccotangent

Derivative of inverse function (f⁻¹)'(x)

1 / f'(f⁻¹(x))

Implicit differentiation Differentiate both sides using chain rule for y and solve for y'

Area of a circle A

πr²

Circumference of a circle C

2πr

Volume of a sphere V

(4/3)πr³

Surface area of a sphere A

4πr²

Volume of a cylinder V

πr²h

Surface area of a cylinder S

2πrh + 2πr²

Volume of a cone V

(1/3)πr²h

Surface area of a cone S

πr(r + √(r² + h²))

Volume of a rectangular pyramid V

(1/3)lwh