Math 160. Final exam practice.
1. Find the limits:
lim
x→3
x−3
x2−x−6lim
x→3−
|x−3|
x−3lim
x→3
x2+ 7x+ 12
x2+ 5x+ 6
2. Use the definition of the derivative to find the derivative for
f(x) = x2+ 3x f(x) = 2
1 + x
3. Let y=x3+x. Give the equation of the secant line through (1, f(1)) and (3, f (3)). Give the
equation of the tangent line at (2, f(2)).
4. Calculate the derivatives of these functions:
f(x) = x
x+ 1 f(x) = (x+ 3)(x2+ 2x+ 1) f(x) = px2+exf(x) = x2ex
f(x) = q1 + √2+3x f(x) = 3 ln x
2 + x2f(x) = e√x2+1 f(x) = ex+e−x
e2x
5. A woman is in a rowboat 3 km offshore. She wants to get to a point on shore 8 km downstream.
She can row at 3 km/hr and walk at 6 km/hr. Where should she land onshore in order to minimize
travel time?
6. We want to fence in a rectangular area with one partition down the middle. One side of the
rectangle is bounded by a river, so it does not require fencing. We have 1000 feet of fence. What
dimensions should we use in order to maximize enclosed area?
7. A restaurant find that if they charge 9 dollars for a meal, they sell 48. If they raise the price to
12 dollars, the number sold drops to 42. It costs 4 dollars to make each dish. How much should
the restaurant charge in order to maximize profits?
8. Use the methods described in chapter 4 to sketch the curves (find first derivative, second
derivative, critical points, inflection points, etc.):
y=x
2x+ 1 y= ln(x2+ 4)
9. Find dy
dx :
x2+xy +y2= 1
