Review for Final Exam

1. All the problems in Exams 1-3.
2. All the reviews.
3. Match the slope fields with their differential equations.
   1. $y^{\prime} = y^{2}$
   2. $y^{\prime} = x-y$
   3. $y^{\prime} = \cos x$
   4. $y^{\prime} = x e^{-x}$
   
   

4. Find the solutions to the following differential equations.

   1. $$\frac{d y}{d x} = 0.5 (y - 200), \quad y(0) = 50 .$$

   2. $$\frac{d y}{d x} = \frac{5y}{x}, \quad y(1) = 3 .$$

   3. $$\frac{d y}{d x} = x e^{y} , \quad y(0)=0 .$$

   4. $$\frac{d y}{d t} = y^{2} (1 + t), \quad y(1)=2 .$$

   5. $$y^{\prime\prime} - 3 y^{\prime} - 4 y = 0, \quad y(0)=1, \ y^{\prime}(0) = 0 .$$

5. Find the Taylor series of the function $${f(x)= x^{2} \arctan(x^{4})}$$ about $x=0$ up to $x^{22}$ .

6. Estimate the value of $${\int_{0}^{1/2} \frac{1}{1+x^{5}} \, d x}$$ by approximating the integrand with a Taylor polynomial of degree $5$ .