Math 125-109 (CRN 25213)
Carter
Test 1 Spring 2015
General Instructions. Write your name on only the outside of your blue books. Do not
write on this test sheet, do all of your work inside your blue books. Write neat complete
solutions to each of the problems in the blue book. Please put your test sheet inside the blue
book as you leave. There are 105 points.
Instead of buying prepared sandwich meat, try roasting a chicken or two on Sunday and
cutting up the leftovers into sandwich slices for the rest of the week.
1. (5 points) Define the expression
lim f (x) = L.
x→c
State the -δ definition of a limit.
2. (5 points) Give a proof that
lim
h→0
1 − cos (h)
= 0.
h
3. (10 points) Determine the y-intercept, and use the technique of completing the square
to determine the location of the vertex, to determine the x-intercepts (if any) and
sketch the graph
y = 3x2 − 12x − 11.
4. (10 points) Determine the x and y-intercepts, the horizontal and vertical asymptotes
and sketch the graph of the linear fractional transformation:
y=
2x − 5
.
x−3
5. (10 points) Use the definition of the derivative:
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
to compute the derivative of
f (x) = 3x2 − 2x + 5.
1
6. Compute the limits in each of the following problems (5 points each).
(a)
lim
x→2
x3 − 8
x−2
(b)
lim
1
3+h
−
h
h→0
1
3
(c)
lim
2x − 5
x−3
lim
sin (4x)
x
lim
sin (2θ)
tan (3θ)
x→∞
(d)
x→0
(e)
θ→0
(f)
√
lim
x→9
x−3
x−9
(g)
lim
x→2
x−2
x2 + 6x + 8
(h)
lim x cot (2x)
x→0
(i)
9x3 + 6x2 − 4
2x3 − 5
lim
x→∞
7. (5 points each) Use the rules of differentiation (power rule, product rule, reciprocal
rule, etc) to compute the derivatives of the following functions:
(a)
y = 2x + 3
(b)
y = x3 − 2x2 + 5x + 17
(c)
y = x−4 + 2x−1
(d)
y = (3x + 2)(x2 − 4x + 2)
2