Calculus Final Exam Review

1. Calculus Final Exam Review By: Bryant Nelson

2. Common Trigonometric Values

3. sin(h) h cos(h) - 1 h lim h0 lim h0 = ? = ? Special Trigonometric Limits 1 0

4. f(x+Δx) – f(x) Δx lim Δx0 Differentiation Rules Definition of a Derivative: f’(x) =

5. Differentiation Rules cont. Product Rule: (f·g)’ = f’·g + f·g’ Quotient Rule: (f/g)’ = (f’·g - f·g’)/g2 Natural Log Rule: d/dx(ln(u)) = (1/u)·du/dx Exponential Rules: d/dx(℮u) = (℮u)·du/dx · d/dx(bu) = (bu) ·ln(b)·du/dx

6. Differentiation Rules cont. Trigonometric Rules d/dx(sin(u)) = (cos(u))du/dx d/dx(cos(u)) = (-sin(u))du/dx d/dx(tan(u)) = (sec2(u))du/dx d/dx(cot(u)) = (-csc2(u))du/dx d/dx(sec(u)) = (sec(u)tan(u))du/dx d/dx(csc(u)) = (-csc(u)cot(u))du/dx

7. Differentiation Rules cont. Inverse Trigonometric Rules d/dx(sin-1(u)) = (1/(√1-u2))du/dx d/dx(cos-1(u)) = (-1/(√1-u2))du/dx d/dx(tan-1(u)) = (1/(1+u2))du/dx d/dx(cot-1(u)) = (-1/(1+u2))du/dx d/dx(sec-1(u)) = (1/(|u|·√u2-1))du/dx d/dx(csc -1(u)) = (-1/(|u|·√u2-1))du/dx

8. Integration Rules Power Rules: ∫(un·du) = (un+1)/(n+1) +C; only while n ≠ -1 ∫(u-1·du) = ln(|u|) +C Exponential Rules: ∫(℮u·du) = ℮u +C ∫(bu·du) = bu/ln(b) - u +C, b≠1 Logarithmic Rule: ∫(ln(u)·du) = u·ln(u) - u +C, u>0

9. Integration Rules cont. Trigonometric Rules ∫(sin(u)·du) = -cos(u) + C ∫(cos(u) ·du) = sin(u) + C ∫(tan(u) ·du) = -ln(|cos(u)|) + C ∫(cot(u) ·du) = ln(|sin(u)|) + C ∫(sec(u) ·du) = ln(|sec(u) + tan(u)|) + C ∫(csc(u) ·du) = ln(|csc(u) + cot(u)|) + C

10. Integration Rules cont. Trigonometric Rules cont. ∫(sec2(u) ·du) = tan(u) + C ∫(csc2(u) ·du) = -cot(u) + C ∫(sec(u)tan(u) ·du) = sec(u) + C ∫(csc(u)cot(u) ·du) = -csc(u) + C

11. n n n n n n ∑ ∑ ∑ ∑ ∑ ∑ c·ak = k2 = ak k3 = k = 1 = K=1 K=1 K=1 K=1 K=1 K=1 c· Summation Formulas n (n(n+1))/2 (n(n+1)(2n+1))/6 (n2(n+1)2)/4