Midterm 1.
Math 106 (23.04.06)
1. For which value of bis the function
g(x) =
x+ 2,if x≤0,
bsin(2x)
x,if x > 0
continuous at the point x= 0.
2. Use the l’Hopital’s rule to find the limit
lim
x→0
sin x−x
x3
3. State the Rolle’s theorem and use it to show that the function f(x) =
x4−4x+ 1 has exactly two zeros on the real line.
4. Sketch the graph of the function
f(x) = x
x−2.
Indicate any horizontal or vertical asymptotes and write the equation of
the tangent line to this graph at the point (3,3).
5. Find the extreme values of the function and where they occur
a)f(x) = 2x2−8x+ 9, b)g(x) = x+4
x,
c) find absolute minimum of g(x) for 0 < x < ∞
6. Find antiderivatives of the following functions
a) sin(2x), b)1
x4c) cos2(3x)