1. Find the following derivatives.
(a) y=4x^5-x*\sqrt{x}, dy/dx=?
(b) y= (2+e^(x+3))/(x^2), dy/dx=?
(c) f(x)=(-x+4)*(x^4+1/x), f '(x)=?
2. Use the limit defn of f'(x) to verify that f'(2)=11 if f(x)=x^3-x.
3. Find the equation of the tangent line to the graph y=x^2+1/x at (1,2).
4. Find the instantaneous velocity at t=4 of a particle whose position at time t is s(t)= 2t^3-t. What is the average velocity on the interval [4,6]?
5. Find the lim_{t->\infty} (t^3+\sqrt(t))/(t+sin(t)-t^3)
6. True of False (You must give at least one reason for your answer in a complete sentence. An answer alone is worth zero.)
If f(x)>0 for all x and is differentiable for all x, then f '(x)>0 for all x.
1. True of False (You must give at least one reason for your answer in a complete sentence. An answer alone is worth zero.)
If f(x) and g(x) are defined for all x but both not continuous at x=0, then f(x)+g(x) is not continuous at x=0.
2. Show that there is some positive number x such that x^8-x-1=0.
3. Find f '(x) if
(a) f(x)= (e^x+1)/(e^x-1)
(b) f(x)=c*x^(1/4)-2, where c is a constant,
4. Does f '(2) exist if f(x)=[x], the greatest integer function?
5. What is lim_{t->1} (t^3-1)/(t-1)? Is this related to a derivative of a well-known function? What is the equation of the tangent line to the graph of y=x^3 at the point (1,1)?
6. Find lim_{x->\infty} (7+x^5-2e^x)/(x^10+e^(2x)).
7. Draw a graph of a function f(x) whose domain is all real numbers, f(x) is continuous everywhere except for x=1 and x=2, f(x) is differentiable everywhere except x=1,x=2 and x=3. Explain why your graph has this required properties.
1. True or False? A function which is continuous at x=a is always differentiable at x=a.
2. Let f(x)= 2x-3 for x>1, and =x^2-2 for 1>=x. Is f(x) continuous at x=1? Is f(x) differentiable at x=1?
3. Find these limits
(a) lim_{h->0} h^2*(cos(1/h)+1/h)
(b) lim_{x->\infty} ((x+\sqrt(x^5+4))/(x^3-x)
(c) lim_{x->2} (x^3-4x)/(x-2).
4. Using the derivative rules find f '(x) (do not simplify)
(a) f(x)= (x^4-2x)/(e^x+x)
(b) f(x)= x*e^x.
5. Water drips from a large tank so that the volume (in gallons) inside the tank is V(t)= 100*(1-t/10)^2, for t between 0 and 10 minutes. Find the instantaneous rate of change of the volume at t=2 minutes.
6. Find the equation of the tangent line to the curve x*y=6 at the point (2,3).