Key to Practice Midterm I

Problem 1

1. True, if dy/dx = g(x)/f(y) then f(y)dy = g(x)dx
   

2. False, 1/x is not continuous at x = 0.
   

3. True, f(g(x)) = x so its derivative is 1.
   

Problem 2

1. (sinx ln(sinx) + cos²x csc x)e^{cosx ln(sinx)}
   

2. 0
   

3. p^x lnp + pxp^-1 + (1 + ln x)x
   

Problem 3

1. ln10 ln(log x) + C
   

2. e^sinx + C
   

3. 2/5(1 - e^x)^5/2 - 2/3 (1 - e^x)^3/2 + C
   

Problem 4

1. Let x = the concentration of CO₂ then
           dx/dt = kx
   

2. x(t) = 281 e^0.0025t
   

3. 408 ppm
   

Problem 5

sqrt(2)

Problem 6

y=(3x³ + cx³)^1/3