Math 125 (CRN 10240,11371,15269) Carter Test 1 Fall 2013
General Instructions: Write your name on only the outside of your blue book. Put your
test paper inside your blue book as you leave. Solve each of the following problems (110
total points). Point values are indicated on the problems. When walking on campus, avoid
walking in the bike lanes.
1. (5 points) Let f : (a, b) → R denote a real valued function that is defined on the open
interval (a, b) = {x ∈ R : a < x < b}. Define the phrase y = f (x) is continuous at
x = c where c ∈ (a, b).
2. (5 points) Give a proof that
lim+
h→0
sin (h)
= 1.
h
3. (5 points) Use the limit rules (LK, LΣ, LΠ, LQ) to prove the product rule for derivatives:
(f · g)0 (x) = (f 0 (x)) · g(x) + f (x) · (g 0 (x)).
Hint: (1) What is the Newton quotient for f · g? (2) Meanwhile,
0 = −f (x) · g(x + h) + f (x) · g(x + h).
4. (10 points) Sketch the graph of
f (x) = 2 sin (x − π/6)
over two full periods.
5. (10 points) Sketch the graph of the linear fractional transformation
f (x) =
x+3
.
x−4
6. (5 points) Use the properties of exponentials and logarithms to simplify the expression
2
eln (x ) .
1
7. Use the rules for computing the limit for each of the following problems (10 points
each).
(a)
x2 − 7x + 10
x→2 x2 − 3x + 2
lim
(b)
tan (2x)
x→0 sin (x)
lim
(c)
lim
1
(3+x)
−
1
3
x
x→0
(d)
3x2 − 2x + 5
x→∞ 2x2 + 3x − 7
lim
8. (10 points) In the figure below, a rectangle is inscribed below the right triangle with
vertices (−1, 0), (0, 1), and (1, 0).
(a) Determine the y-coordinate of the point P.
(b) Express the area of the rectangle in terms of the variable x.
Hint: What is the equation of the line on the right?
y
(1,1)
P( x, ?)
x
0
(-1,0)
x
(1,0)
9. (10 points) Use the definition of the derivative, f 0 (x) = limh→0
(x)
), to compute the derivative of the function
limz→x f (z)−f
z−x
√
f (x) = x.
f (x+h)−f (x)
h
(or f 0 (x) =
10. (10 points) Use the definition of the derivative (as given above) to compute the equation
of the line tangent to the curve
y = x3
at the point (2, 8).
2