Introduction :
Math is used throughout the whole world that has fundamental tool in various fields that include natural science, engineering, medicine, and the social sciences. Mathematics is the learning of quantity, arrangement, space, and change. Math seeks out patterns that originate the new conjecture, and ascertain truth by precise deduction from properly selected axioms and definitions.
Example Problems for Problem Based Learning Math:
Problem based learning math – Example: 1
Find a function that has choral on the slip between the lines `y=-x+3, y=-x-3` that takes the values -50 and 10 on the lower and upper lines.
Solution:
Guess `\phi(x,y) = Ax + By + C`
Find the values of `A, B, C.`
`\phi(3,0) = 3A+C=10`
`\phi(-3,0) = -3A+C=-50`
`10 = -50 + 6A`
`A = 10`
`\phi(0,3) = 3B+C=10`
`\phi(0,-3) = -3B+C=-50`
`B = 10`
`\phi(3,0) = 30 + C = 10, C = -20`
The solution is
`\phi(x,y) = 10x + 10y - 20`
Problem based learning math – Example: 2
Find the partial fraction decomposition of `\frac{4z+4}{z(z-1)(z-2)^2}.`
Solution:
`\frac{4z+4}{z(z-1)(z-2)^2} = \frac{A}{z} + \frac{B}{z-1} + \frac{C}{z-2} + \frac{D}{(z-2)^2}`
`4z+4=A(z-1)(z-2)^2 + Bz(z-2)^2 + Cz(z-1)(z-2) + D z(z-1)`
Plug in z=0 to get A=-1
Plug in z=1 to get B=8
Plug in z=2 to get D=6
Differentiate both sides of the equation once with respect to z and plug in z=2 to get C=-7.
Finally
`\frac{4z+4}{z(z-1)(z-2)^2} = \frac{-1}{z} + \frac{8}{z-1} - \frac{7}{z-2} + \frac{6}{(z-2)^2}`
Problem based learning math – Example: 3
If `u(x,y) = e^x\sin y` find `f(x,y) = u(x,y) + i v(x,y)` and check if it satisfies the Cauchy-Riemann equations.
Solution:
The Cauchy-Riemann equations are `u_x=v_y, v_x=-u_y.`
`u_x = e^x\sin y, u_y = e^x\cos y`
`v_y = e^x\sin y`
`v =-e^x\cos y+g(x)`
`v_x =-e^x\cos y+g'(x)`
For the CR equations to hold, we must have `g'(x)=0` so that `g(x)=c\isin{R}.`
`f(x,y) = e^x\sin y + i(-e^x\cos y + c)`
`=e^x(\sin y-i\cos y) + ic`
`=-i e^x(\cos y + \frac{1}{-i}\sin y) + ic`
` =-i e^x e^{iy} + ic = -i e^z+ic`
Please express your views of this topic polynomial word problems by commenting on blog.
Practice Problems for Problem Based Learning Math:
1. Show that if `\phi(x,y)` is harmonic then `\phi_x - i \phi_y` is analytic.
2. Find a function that is choral on the vertical slip from x = 1 to 2 and equals 20 and 30 at x = 1 and 2.
`Answer: \phi(x,y) = 10x+10 `
Math is used throughout the whole world that has fundamental tool in various fields that include natural science, engineering, medicine, and the social sciences. Mathematics is the learning of quantity, arrangement, space, and change. Math seeks out patterns that originate the new conjecture, and ascertain truth by precise deduction from properly selected axioms and definitions.
Example Problems for Problem Based Learning Math:
Problem based learning math – Example: 1
Find a function that has choral on the slip between the lines `y=-x+3, y=-x-3` that takes the values -50 and 10 on the lower and upper lines.
Solution:
Guess `\phi(x,y) = Ax + By + C`
Find the values of `A, B, C.`
`\phi(3,0) = 3A+C=10`
`\phi(-3,0) = -3A+C=-50`
`10 = -50 + 6A`
`A = 10`
`\phi(0,3) = 3B+C=10`
`\phi(0,-3) = -3B+C=-50`
`B = 10`
`\phi(3,0) = 30 + C = 10, C = -20`
The solution is
`\phi(x,y) = 10x + 10y - 20`
Problem based learning math – Example: 2
Find the partial fraction decomposition of `\frac{4z+4}{z(z-1)(z-2)^2}.`
Solution:
`\frac{4z+4}{z(z-1)(z-2)^2} = \frac{A}{z} + \frac{B}{z-1} + \frac{C}{z-2} + \frac{D}{(z-2)^2}`
`4z+4=A(z-1)(z-2)^2 + Bz(z-2)^2 + Cz(z-1)(z-2) + D z(z-1)`
Plug in z=0 to get A=-1
Plug in z=1 to get B=8
Plug in z=2 to get D=6
Differentiate both sides of the equation once with respect to z and plug in z=2 to get C=-7.
Finally
`\frac{4z+4}{z(z-1)(z-2)^2} = \frac{-1}{z} + \frac{8}{z-1} - \frac{7}{z-2} + \frac{6}{(z-2)^2}`
Problem based learning math – Example: 3
If `u(x,y) = e^x\sin y` find `f(x,y) = u(x,y) + i v(x,y)` and check if it satisfies the Cauchy-Riemann equations.
Solution:
The Cauchy-Riemann equations are `u_x=v_y, v_x=-u_y.`
`u_x = e^x\sin y, u_y = e^x\cos y`
`v_y = e^x\sin y`
`v =-e^x\cos y+g(x)`
`v_x =-e^x\cos y+g'(x)`
For the CR equations to hold, we must have `g'(x)=0` so that `g(x)=c\isin{R}.`
`f(x,y) = e^x\sin y + i(-e^x\cos y + c)`
`=e^x(\sin y-i\cos y) + ic`
`=-i e^x(\cos y + \frac{1}{-i}\sin y) + ic`
` =-i e^x e^{iy} + ic = -i e^z+ic`
Please express your views of this topic polynomial word problems by commenting on blog.
Practice Problems for Problem Based Learning Math:
1. Show that if `\phi(x,y)` is harmonic then `\phi_x - i \phi_y` is analytic.
2. Find a function that is choral on the vertical slip from x = 1 to 2 and equals 20 and 30 at x = 1 and 2.
`Answer: \phi(x,y) = 10x+10 `
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