1. Evaluate the following limits analytically. If the limit does not exits, determine the onesided limits.
x sin x + x2
x→0
x
2
x +x−2
lim
x→4 x2 − 2x − 8
sin 4θ
limπ
θ→ 2
θ
√
4x2 + 3x
lim
x→∞ 6x4 − 2
|t|
lim sin t
t→0
t
(a) lim
(b)
(c)
(d)
(e)
2. (a) State the formal definition for the limit of f (x) as x approaches c.
(b) Prove rigorously lim 3x − 1 = 11
x→4
(c) Find δ satisfying the definition of the limit for
ǫ = 0.01 and lim 3x − 1 = 11
x→4
(d) Consider the limit lim 3x − 1. If ǫ = 1, then δ = 0.25 satisfies the defintion of the
x→4
limit. That is, when a ∈ (3.75, 4.25) then f (a) = 3a − 1 is in what interval?
3. (a) Define f (x) is a “continuous” at x = c.
(b) Find values of a and b so that f (x) is continuous.
 2
 x +2 x<1
f (x) =
3a
x=1

ax − b x > 1
(c) Discuss the continuity of g(x) =
tan x
on the interval [0, π].
x−1
4. (a) State the Intermediate Value Theorem.
(b) Use the Intermediate Value Theorem to show that f (x) = x3 + 3x − 7 has a zero on
[1,3].
(c) Find an interval of length 1 that contains a zero of f (x) = x3 + 3x − 7.
5. Sketch the graph of a function f that satisfies the following conditions:
lim f (x) = −1,
x→−2
lim f (x) = −2,
x→1−
lim f (x) = 1
x→1+
f (1) is undefined
6. Below is the graph of the function f (x).
(a) Use the graph of f (x) to evaluate the following limits if they exist.
lim f (x) =
lim f (x) =
x→−2
x→1
lim f (x) =
lim f (x) =
x→−1−
x→1−
lim f (x) =
lim f (x) =
x→−1+
x→1+
lim f (x) =
lim f (x) =
x→−1
x→3
(b) State the points at which f (x) is not continuous.
y
6
5
4
3
2
bc
1
x
−2
−1
1
2
3
7. (a) State a formal definition of “f ′ (x) is the derivative of f (x).”
(b) Using the definition of derivative determine
d
[7x2 + 3]
dx
(c) Using the definition of derivative determine
d
[3x2 + 7]
dx
8. Use the limit definition to compute f ′ (a) and find the equation of the tangent line.
(a) f (x) = x + x−1 , a = 4
(b) f (x) = t − 2t2 , a = 3
1
, a = −2
(c) f (x) =
x+3
9. Compute the following derivatives.
d 4
x |x=−2
dx
d 14/3
x
(b)
dx
√
(c) f ′ (4), where f (x) = 5x − 32 x
(a)
10. Calculate the derivative.
(a) f (x) = 2x3 − 3x2 + 2x
(b) h(x) = π 2
(c) P (s) = (4s − 3)2
11. If possible, use the given graph of f (x) to evaluate the following derivative. If the derivative does not exist, state why not.
• f ′ (−1) =
• f ′ (2) =
• f ′ (1) =
• f ′ (5) =
y
3
2
b
1
x
−2
1
−1
2
3
4
5
6
12. Below is the graph of the function g(x). On the axes provided, sketch the derivative of
g(x).
y
y
g(x)
3
3
2
2
1
1
x
−2
1
−1
−1
2
3
x
−2
1
−1
−1
2
3