Problem Set #2 – Deriving Derivatives
1. Calculate f'(0) when f(x)=x9+x8+4x5-7xx4-3x2+2x+1. Hint: It might be easier to label with the function fx as u(x)v(x) and calculate f'(x) in terms of u and v first.
2. Find constants A and B such that y=Asinx+Bcos(x) satisfies the differential equation y''+ y'- 2y=sin(x).
3. How many tangent lines to the curve y=xx+1 pass through point (1, 2)? At what points do these tangent lines touch this curve (only x-values required)?
4. f4=2 f'4=-3 g4=5 g'4=-1
a. Find the derivative of f(x)g(x) and f(x)/g(x) at x = 4.
b. Calculate h'4 where hx=xg(x).
5. In 1999, the population in the Richmond, VA area was 961,400 and increasing at about 9200 people per year. The average annual income per capita was $30,593 and this average was increasing at about $1,400 per year. Total income in the Richmond area is the population * average annual income. Use the Product Rule to estimate the rate at which the total income was rising in the Richmond area. Explain the meaning of each term in the Product Rule.
6. For what values a and b is line 2x+y=b tangent to parabola y=ax2 when x=2?
7. The following three graphs show f, f’ and f’’. Which one is which? Explain how you determined this.
A. B. C.
8. The following is a table of values for f(x), g(x), f’(x) and g’(x).
x / f(x) / g(x) / f’(x) / g’(x)1 / 3 / 2 / 3 / 6
2 / 1 / 8 / 5 / 7
3 / 2 / 2 / 8 / 11
a. If h(x) = f(g(x)), find h’(1).
b. If k(x) = g(f(x)), find k’(3).
9. Consider the curve given by xy2-x3y=6.
a. Show that dydx=3x2y-y22xy-x3.
b. Find all points on the curves whose x-coordinate is 1, and write an equation for the tangent line at each of these points.
c. Find the x-coordinate of each point on the curve where the tangent line is vertical (yes, the answer isn’t clean).
10. A helicopter is moving up and down according to a law of motion h(t) where h is distance (measured in yards) and t is time (measured in seconds). 0≤t ≤9.
ht=t3-7t2-5t+110
a. What is the velocity of the helicopter at time 2? At time t?
b. At what time(s) is the helicopter not moving?
c. For what time interval(s) is the helicopter rising?
d. What is the lowest height the helicopter ever reaches?
e. What is the acceleration of the helicopter at time 3? At time t?
f. For what time interval(s) is the helicopter slowing down?
11. Find a parabola that passes through the point (-1, 0) and has tangent lines with slopes of 4 and -2 at x= -3 and 1, respectively.
12. What is the equation of the tangent line to the curve corresponding to the given value of the parameter. x=t4+1 y=t3+t t=-1