Math transfer 4/24/25

9 flashcards · Mathematics · by Harrison Holbrook

1. Partial fractions

   Rational functions in lowest terms, ex. p(x)q(x)\frac{p(x)}{q(x)}q(x)p(x)​ , where the degree of q(x)q(x)q(x) is greater than that of p(x)p(x)p(x) , can be factored into simpler rational functions, called partial fractions. These can be trivially integrated using natural logarithms.

2. Partial fraction decomposition when the denominator contains distinct linear factors (Heaviside cover-up method)

   If the denominator contains distinct linear factors, ex. c(x+a)(x+b)\frac{c}{(x+a)(x+b)}(x+a)(x+b)c​ , then the fraction can be decomposed by Setting up fractions in the form A(x+a)+B(x+b),...\frac{A}{(x+a)}+\frac{B}{(x+b)},...(x+a)A​+(x+b)B​,... for each linear factor of the denominator Determine the coefficients of each of these partial fractions In the original fraction, mentally "cover-up" or remove the denominator's linear factor associated with the desired coefficient, ex. (x+a)(x+a)(x+a)…

3. Partial fraction decomposition when the denominator contains a repeated linear factor

   If the denominator contains repeated linear factors, ex. c(x+a)n\frac{c}{(x+a)^n}(x+a)nc​ , then the decomposition of the fraction includes the terms A1(x+a)+A2(x+a)2...An(x+a)n\frac{A_1}{(x+a)}+\frac{A_2}{(x+a)^2}...\frac{A_n}{(x+a)^n}(x+a)A1​​+(x+a)2A2​​...(x+a)nAn​​ , and can be decomposed by Setting up partial fractions as normal (each term of the denominator) in addition to the rule above, ex. ` cx(x+a)n=Ax+B(x+a)+C(x+a)2...D(x+a)n\frac{c}{x(x+a)^n}=\frac{A}{x}+\frac{B}{(x+a)}+\frac{C}{(x+…

4. Partial fraction decomposition when the denominator contains a distinct irreducible quadratic factor (cannot be factored into linear factors)

   If the denominator contains a distinct irreducible quadratic factor ax2+bx+cax^2 + b x + cax2+bx+c , then the decomposition of the fraction includes the term Ax+Bax2+bx+c\frac{Ax+B}{ax^2+bx+c}ax2+bx+cAx+B​ , and can be decomposed by Setting up partial fractions as normal (each term of the denominator) in addition to the rule above, ex. 3xx3−1=3x(x−1)(x2+x+1)=A(x−1)+Bx+Cx2+x+1\frac{3x}{x^3-1}=\frac{3x}{(x-1)(x^2+x+1)}=\frac{A}{(x-1)}+\frac{Bx+C}{x^2+x+1}x3−13x​=(x−1)(x2+x+1)3x​=(x−1)A​+x2+x+1Bx+…

5. Partial fraction decomposition when the denominator contains a repeated irreducible quadratic factor (cannot be factored into linear factors)

   If the denominator contains a repeated irreducible quadratic factor (ax2+bx+c)n(ax^2+bx+c)^n(ax2+bx+c)n , then the decomposition of the fraction includes the terms A1x+B1(ax2+bx+c)+A2x+B2(ax2+bx+c)2+Anx+Bn(ax2+bx+c)n\frac{A_1x+B_1}{(ax^2+bx+c)}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\frac{A_nx+B_n}{(ax^2+bx+c)^n}(ax2+bx+c)A1​x+B1​​+(ax2+bx+c)2A2​x+B2​​+(ax2+bx+c)nAn​x+Bn​​ , and can be decomposed by Setting up partial fractions as normal (each term of the denominator) in addition to the rule above, ex.…

6. Trapezoidal sum (conceptual definition)

   A trapezoidal sum is the average of a left Riemann sum and a right Riemann sum

7. Error in the midpoint Riemann sum

   Error≤(b−a)3M24n2\text{Error} \leq \frac{(b-a)^3M}{24n^2}Error≤24n2(b−a)3M​ , where MMM is the absolute maximum value of ∣f′′(x)∣|f''(x)|∣f′′(x)∣ on the closed interval [a,b][a,b][a,b]

8. Simpson's Rule (Simpson's Method for approximating definite integrals)

   ∫abf(x)dx=b−a3n[f(x0)+4f(x1)+2f(x2)+4f(x3)...2f(xn−2)+4f(xn−1)+2f(xn)]\int_a^b f(x)dx=\frac{b-a}{3n}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)...2f(x_{n-2})+4f(x_{n-1})+2f(x_n)]∫ab​f(x)dx=3nb−a​[f(x0​)+4f(x1​)+2f(x2​)+4f(x3​)...2f(xn−2​)+4f(xn−1​)+2f(xn​)] , where the closed interval [a,b][a,b][a,b] is partitioned into an even number nnn of subintervals [x0,x1],[x1,x2]...[xn−1,xn][x_0, x_1], [x_1,x_2] ... [x_{n-1}, x_n][x0​,x1​],[x1​,x2​]...[xn−1​,xn​] of width b−an\frac{b-a}{n}nb−a​

9. Error in Simpson's Rule

   Error≤(b−a)5M180n4\text{Error} \leq \frac{(b-a)^5M}{180n^4}Error≤180n4(b−a)5M​ , where MMM is the absolute maximum value of ∣f(4)(x)∣|f^{(4)}(x)|∣f(4)(x)∣ on the closed interval [a,b][a,b][a,b]