MATH 101 HOMEWORK 1
Due on Wednesday Sept. 15.
For full credit, show all work. Calculators are not allowed.
√
1. (5 marks) Find numbers a and b such that lim
x→0
ax + b − 4
= 1.
x
2. (5 marks) (a) Check that for each value of the parameter m, the line y = mx − (m2 /4)
is tangent to the parabola y = x2 .
(b) If a, b, c are given, find a function f (m) such that for each m the line y = mx + f (m)
is tangent to the parabola y = ax2 + bx + c.
3. (5 marks) Let ABC be a triangle with 6 BAC = 120 degrees and |AB| · |AC| = 1.
(a) Express the length of the angle bisector AD in terms of x = |AB|.
(b) Find the largest possible value of |AD|.
4. (5 marks) Sketch the graph of the function y = e−1/(x+1) . Include the following
information: local minima and maxima, intervals of increase and decrease, intervals of
concavity and inflection points, asymptotes, limits at infinity.
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