Calculus Exam Formula Sheet PDF

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Calculus Exam Formula Sheet Chapter 1: Remember: to simplify numerator (conjugate if needed), Average rate of Change: # %# 𝑚 = ' $%'& $

&

The Difference Quotient: 𝑓(𝑥 + ℎ) − 𝑓(𝑥) 𝑓′(𝑥) = lim 0→2 ℎ Limits: A function exists if: - lim7 𝑓 (𝑥)8𝑒𝑥𝑖𝑠𝑡𝑠 '→6

- lim= 𝑓 (𝑥)8𝑒𝑥𝑖𝑠𝑡𝑠 '→6

Continuous if : 𝑓(𝑥) - lim= 𝑓 (𝑥)8 = lim 7 '→6

'→6

Jump Discontinuity= lim= 𝑓(𝑥 )8 ≠ lim7 𝑓 (𝑥 ) '→6 '→6 Substitution Question: @ 𝑥 (𝑎𝑛𝑑8𝑜𝑡ℎ𝑒𝑟8𝑣𝑎𝑙𝑢𝑒) Let u= √ *Isolate for x *Sub x→x value into let statement to get the u→value u→ @√𝑢𝑠𝑒8𝑡ℎ𝑒8𝑥 → #8ℎ𝑒𝑟𝑒 Rewrite the limit with u on top and isolated x on bottom 𝑎J + 𝑏J = (a + b)(𝑎N − 𝑎𝑏 + 𝑏N ) 𝑎J − 𝑏J = (a − b)(𝑎N + 𝑎𝑏 + 𝑏N ) Cancel out top and bottom Rewrite and sub the u→ into8u Substitution Question (multiple): S @ When you have √ 8𝑎𝑛𝑑8 √ , let u=√ @ Do *’s. Do 2 formulas’, one with √ and another with √ . Within the roots, place @ your u6. The √ will get the u6 to become 3 u and the other will become u2. Now plus the new u values for x into the original. Find a common denominator and multiply by the reciprocal. Use cube root formulas to cancel Chapter 2: Chain Rule: 𝑓 (𝑥) = 𝑓(𝑔(𝑥)), 𝑓 V (𝑥 ) = 𝑓′(𝑔 (𝑥)) × 𝑔′(𝑥) Product Rule: 𝑓(𝑥) = 𝑓 (𝑥) × 𝑔(𝑥 ), 𝑓V (𝑥) = 𝑔 (𝑥)𝑓 V(𝑥 ) + 𝑓(𝑥)𝑔′(𝑥) 8 Quotient Rule:

𝑓(𝑥 ) V , 𝑓 (𝑥) 𝑔(𝑥 ) V 𝑔(𝑥)𝑓 ( 𝑥) − 𝑓(𝑥)𝑔V(𝑥) = [𝑔(𝑥 )] N Chain and Product Rule Rule:: 𝑓( 𝑥) Q= 8Q(x)g(x ) 𝑔 (𝑥) = f(x)Q’(x)g(x) + g’(x)Q(x) = f’(x)Q’ (x)g( x) = f’(x ) − g’ (x)Q(x )Q’(x )g (x ) 𝑓(𝑥) ` 8Q’ (x) g(x) = f’(x) − g’(x) _8 𝑔(𝑥) = 𝑑 − 𝑔28on8bottom8then8flip 𝑓(𝑥) =

Double Chain Rule Questi Question: on:e

'%f

('$ gf)$

J

h

distribute the power to the numerator and denominator before quotient rule

Derivatives of Functi Functions: ons: 𝑓(𝑥) = 𝑠𝑖𝑛𝑢888𝑓 V (𝑥) = 𝑐𝑜𝑠𝑢 × 8𝑢′ 𝑓(𝑥) = 𝑐𝑜𝑠𝑢888𝑓 V (𝑥) = −𝑠𝑖𝑛𝑢 × 8𝑢′ 𝑓(𝑥) = 𝑡𝑎𝑛𝑢888𝑓 V( 𝑥) = 𝑠𝑒𝑐N 𝑢 × 8𝑢′ 𝑓(𝑥) = 𝑠𝑒𝑐𝑢888𝑓 V (𝑥) = 𝑠𝑒𝑐𝑢 × 𝑡𝑎𝑛𝑢8 × 8𝑢′ 𝑓(𝑥) = 𝑐𝑠𝑐𝑢888𝑓 V (𝑥) = −𝑐𝑠𝑐𝑢 × 8𝑐𝑜𝑡𝑢8 × 𝑢’ 𝑓(𝑥) = 𝑐𝑜𝑡𝑢888𝑓 V (𝑥) = −𝑐𝑠𝑐 N 𝑢 × 8𝑐𝑜𝑡𝑢8 × 𝑢’ 𝑓 (𝑥) = 𝑏j 88𝑓 V (𝑥) = 𝑏j × 𝑙𝑛𝑏 × 𝑢′ 𝑓( 𝑥) = 𝑒 j 888𝑓V(𝑥) = 𝑒 j × 8𝑢′ 1 𝑓(𝑥) = 𝑙𝑛𝑢888𝑓 V( 𝑥) = × 8𝑢′ 𝑢 1 8 × 𝑢′ 𝑓( 𝑥) = 𝑙𝑜𝑔6 j 88𝑓V (𝑥) = 𝑢 × 𝑙𝑛6 2 Note: when you have a cot x, change it to (cotx)2 and use chain rule

• Plug x into original + solve for y Critical Point Points: s: Same as max/min (see above) Points of Inf Inflection: lection: Same as max/min using second derivative 2nd Derivative Tes Test: t: • Plug x value into critical point of second derivative If you get a + value, it’s a min (local) If you get a – value, it’s a max (local) Absolute Max/Min • Find local max/min • Evaluate ends of graphs YY-Int. Int. • Make x=0 XX-Int. Int. • Make y=0 Domain: 𝑋 ∈ 𝑅 Range: Y ∈ 𝑅 Chapter 4: Rectangular Prism SA=2lw+2lh+2wh V=lwh Cylinder SA=2𝜋𝑟N + 2𝜋𝑟ℎ V=𝜋𝑟N ℎ pqr @

Sphere SA=4𝜋𝑟N V = J Rectangle P =2l+2w A=lw s×0 Triangle P =a+b+c A= N Circle C=2𝜋𝑟 A=𝜋𝑟N Let x=radius let diameter=2x Solving Exponents with a (x or t): • Isolate Base • Ln both sides • Power rule • Isolate variable Shaded Area=area of square-area of circle Half Life Question: A(final), P(initial), R(rate), t(specific time), v

Remember to distribute the powers to Chapter 3: Velocity= 1st derivative Acceleration=2nd derivative C ( x) = Cost Function C ¢ (x ) = Marginal Cost Function p( x) =Price or Demand Function R( x) = xp( x) = Revenue Function R ¢( x) = Marginal Revenue Function P (x ) = R (x ) - C (x ) = Profit Function C ¢(x ) = R ¢(x ) = Maximum Profit C( x) = Ave Cost Function x C (x ) = Average Cost Min C ¢(x ) = x

c (x ) =

Money Quest Question: ion: Make two equations: X=units, p=price X=sb+/-#change in sb(n) P=each+/-#change in price(n) -use x equation and isolate for n -sub into p equation -find revenue [R(x)=x(p)] (if m, derive and plug x value in) Max/Min Max/Min:: • Take derivative • Make derivative = 0 • Solve for x

h(half-life) Formula: 𝐴 = 𝑃(𝑅) w Use solving exponents rules v

Half-life means, 𝐴 = 𝑃(0.5)w Rate=derivative – Q: at what rate is the mass decaying after __(plug into t) 𝑏j → 𝑏j × 𝑙𝑛𝑏 × 𝑢′ Special Triangles: { SOH - csc8 | {

CAH - sec8 TOA - cot 8

} } |

Log Rules:

𝑙𝑜𝑔6 𝑎 = 1 𝑙𝑜𝑔6 𝑏 ~ = 𝑐 𝑙𝑜𝑔6 𝑏 𝑏 𝑙𝑜𝑔6 • € = 𝑙𝑜𝑔6 𝑏 − 𝑙𝑜𝑔6 𝑐 𝑐 log 6( 𝑏 × 𝑐) = 𝑙𝑜𝑔6 𝑏 + 𝑙𝑜𝑔6 𝑐

Ln Rules:

lne=18 𝑙𝑛𝑒 6 = 𝑎 ( 𝑙𝑛𝑒)8 ln(a× 𝑏) = 𝑙𝑛𝑎 + 𝑙𝑛𝑏 8 𝑎 ln e h = 𝑙𝑛𝑎 − 𝑙𝑛𝑏 8 𝑏...