Math 1251 Practice Exam 2

1. Find the following derivatives and limits.

(a) y=x*ln(-x^2+sin(2x)), dy/dx=?

(b) y=e^(7x+arctan(3x)) , dy/dx=?

(c) log(y)+x*sin(y)=x^5+2, dy/dx=?

(d) lim_{t->-infty} (t^10+e^t)/(e^(2t)+log(-t)-t^8)

(e) lim_{x->0}(1-cos(2x))/(x*arcsin(x)) .

2. Find a function f(x) such that f'(x)=-5*f(x) for all x and f(0)=12.

3. Find the absolute minimum and absolute maximum value of the function f(x)=x*ln(x) on the interval [1/10,4].

4. The area A of an ellipse x^2/a^2+y^2/b^2=1 is known to be A=\pi*a*b. Suppose that an ellipse is growing in such a way the dA/dt=10. Find da/dt when a=3, b=5, and db/dt=4.

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

(a) e^x is an increasing function of x for all x.

(b) If f'(x)=0, then x must be either a local max or a local min for f(x).

(c) If f''(x)>0 for all real x, then f(x) is an increasing function of x for all x.