1. Find the following derivatives and limits.
(a) y=x*ln(-x2+sin(2x)), dy/dx=?
(b) y=e7x+arctan(3x) , dy/dx=?
(c) log(y)+x*sin(y)=x5+2, dy/dx=?
(d) lim_{t->0} sin(4t2)/(t sin(3t)).
2. Find the absolute minimum and absolute maximum value of the function f(x)=x*ln(x) on the interval [1/10,4].
3. The area A of an ellipse x2/a2+y2/b2=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.
4. Find the best linear approximation to \sqrt(1+h)+\cuberoot(8+h) if h is small.
5. True of False (If true give a proof, if false give a counterexample. An answer alone is worth zero.)
(a) ex 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.
(d) A function which is increasing on [0,1] is always concave upward on [0,1].
1. Find the following derivatives and limits.
(a) y=xln(5x), dy/dx=?
(b) y=eg(x) , dy/dx=?
(c) log(y)+x*cos(y)=2, dy/dx=?
(d) lim_{x->0} x*tan(7x2)/sin(2x3).
2. By taking the logarithmic derivative of f(x)*g(x), prove the product rule.
3. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 (cm)3/sec, how fast is the water level rising when the water is 5 cm deep?
4. Suppose we know that f(x) is differentiable everywhere, f(0)=0, and |f '(x)| is at most 1. Prove that |f(x)| is at most |x|.
5. Find the local max-mins and inflection points of the function f(x)=x^3-x^2, and use the second derivative test to verify your claims about the local max-mins.
6. True of False (If true give a proof, if false give a counterexample. An answer alone is worth zero.)
(a) A continuous function on the open interval (0,1) always has an absolute maximum.
(b) The best linear approximation to cos(h)+\sqrt(4+h) for small h is 3+h/4.
(c) The derivative of e7 is either 7*e6 or e7 or 7*e7.
(d) The nth derivative of (1+x)n is n!.
(e) The function f(x)= x*ln(x)-x has no local minima on the interval [1/2,2].
7. Two rare Galapagos tortoises, Darwin and Huxley, engage in a playful race. They start 5 feet apart, and Darwin walks north at 1 in/sec, while Huxley walks east at 2 in/sec. What is the rate of change of the distance between them 10 minutes later?