Find the equation of the line tangent to the curve fx x3 -

Question

1. Find the following limits, if they exist. If a limit does not exist, write DNE and explain why.

a) lim_x→0 (x-1)²/x²

b) lim_x→0 (2x² - 3x + 5/-x³ - 8x² + 2x - 6)

c) lim_x→0 (x-9/√x-3)

d) lim_x→0|x+2|/x+2

2. Find the equation of the line tangent to the curve f(x) = x³ - 2x² + 3x - 8 when x = 1.

3. Use implicit differentiation to find dy/dx for e^xy - 2 = (5x + 3)².

4. For the function f(x) = 1/x²+1, find:

(a) the domain

(b) the intercepts

(c) the equations for any asymptotes

(d) the critical points

(e) the intervals of increase or decrease

(f) local extrema

(g) the intervals of concave up or concave down

(h) inflection points Then sketch the graph. (4 points)

5. A farmer wants to enclose a rectangular field with 600 yd of fencing that borders a straight river. He needs no fencing along the river. What are the dimensions of the field with largest area? Use Lagrange Multipliers to solve this problem.

6. Find all local extrema or saddle points for f(x, y) = x² - 2xy² + ½y².  

7. Integrate:

a) ∫(3x³ - 5x + 2 - 8/x - e^x)dx

b) ∫(ln(3x)/x)dx

c) 1∫2((ln(x))²/x³)dx

8. Find the area enclosed by y = x² + 4x + 2 and y = -1.