Wk10 0B2 Tutorial (Improper Integrals and Isoclines)

Which of these integrals are improper integrals?

• $\int_{-\infty}^1\ x^{4\ }dx$

• $\text{\displaystyle}\int_{-7}^5\ \frac{x+2}{x-5}dx\$

• $\int_{-7}^{10}\ \frac{x+2}{x-5}dx\$

• $\int_{-7}^{-1}\ \frac{x+2}{x-5}dx\$

• $\int_{-1}^{\infty}\ x^{4\ }dx$

Why is $\int_{-7}^{10}\ \frac{x+2}{x-5}dx\$  an improper integral?

• The integrand  $\frac{x+2}{x-5}$  is improper.

• The limits of the integral are infinite.

• The integrand  $\frac{x+2}{x-5}$  has a discontinuity at one of the limits of integration.

• The integrand  $\frac{x+2}{x-5}$  has a discontinuity outside the interval of integration.

• The integrand  $\frac{x+2}{x-5}$  has a discontinuity inside the interval of integration.

What is $\int_2^5\ \frac{1}{\sqrt{5-x}}dx$ ?

•  $\infty$  

•  $2\sqrt{3}$  

•  $-\infty$  

•  $\sqrt{3}$  

• 0

• 1/2

• -1/2

• 0

• 1

• The inclination of solutions of the ODE.

• Graph sketches of solutions of the ODE.

• Functions y(x) satisfying the equation F(x,y)=c for some constant c.

• The graph of the equation F(x,y)=c for some constant c.

Why can isoclines be useful?

• Can be used to extend the direction to problematic points discussed in question

• We can use the isocline to derive a solution of the ODE.

• Isoclines are solutions to the ODE.

• We can use them to obtain a more accurate direction field.