Math 1251 Practice Exam 2

1. Find the following derivatives and limits.

(a) y=e^(4x+arcsin(x^2)), dy/dx=?

(b) y= x ln(x^2+4), dy/dx=?

(c) y^4+cos(xy)=4+e^(2x), dy/dx=?

(d) lim_{t->infty} (t^10+e^t)/(e^(2t)+log(t))

(e) lim_{x->0} arcsin(x)/x.

2. Under certain conditions, the pressure P and the volume V of air satisfy P*V^{1.4}=C, where C is a constant. Find dV/dt if P=5, V=10, and dP/dt=2.

3. Find the absolute minimum and absolute maximum value of the function f(x)=x e^(-x) on the interval [1/2,5].

4. True of False (If true give a proof, if false give a counterexample. An answer alone is worth zero.)

(a) If f'(x)>0 for all x and f(0)=2, then f(x)>0 for all x>0.

(b) If f'(x)>0 on the interval [1,3], then f''(x)>0 on the interval [1,3].

(c) If f''(x) is continuous on [0,1] with f'(c)=0 and f''(c)>0 for some c in [0,1], then f has a local minima at c.