Math 1210-1
Review 1
1. Find the slope of the line containing the indicated two points.
(a) (0, 2) and (9, −4)
(b) (5, −3) and (−8, 13)
2. Find the equation of the line passing through the indicated point with the given slope.
(a) Slope is -3 and the point is (7, 3)
(b) Slope is
2
7
and the point is (5, 9)
3. Find the equations of the lines passing through the point (2, 1) with one line perpendicular to 4x−3y = 9
and the other line parallel to the given line.
4. If f (x) = 4x3 − 8x + 7, find the indicated values.
(a) f (1)
(b) f (−1)
(c) f ( 1t )
(d) f (x + h)
f (x + h) − f (x)
(e)
h
5. Determine which of the following are functions.
(a)
y
†
x
⊙
♦
♠
♥
♦
♥
♥
△
(b)
x
♭
♮
♯
♮
♠
y
♠
♥
♣
♦
♠
(c)
x
⊙
⋆
⊖
∗
⊘
y
⊕
⊘
⊗
⋄
⊕
6. State whether each function is even, odd, or neither.
x−1
x+1
(b) g(x) = 5x2 + 6
(a) f (x) =
(c) h(x) = x3 − 2x
(d) k(x) = sin x cos x
7. Given f (x) = 3x2 − 5x + 8 and g(x) =
(a) (f − g)(x)
1
, find each value:
x−1
(b) (f · g)(x)
(c) (f /g)(3)
(d) (f ◦ g)(x)
(e) (g ◦ g)(x)
(f) (f 2 )(x)
8. What is are the period, amplitude, and vertical and horizontal shifts of y = 3 sin(x − π6 ) + 2?
9. Find the indicated limits.
(a) lim (x − 3)
x→5
(b) lim (x2 + 3x − 4)
x→−3
x2 − 4x − 21
x→−3
x+3
1/3
7t + 5
(d) lim 2
t→1 t − 2t + 8
1
(e) lim x sin
x→0
x
(c) lim
1 − cos2 t
t→0
t
2
sin t
lim
t→0
t
x4 + 3x2
lim
4
x→∞ 7x − 9x3 + 7x − 1
x3 + 1
lim 5
x→∞ x − 1
3x6 − 8
lim
x→∞ 2x3 − 4
x
lim
−
t→2 (x − 2)(x − 1)
x
lim
+
s→2 (x − 2)(x − 1)
(f) lim
(g)
(h)
(i)
(j)
(k)
(l)
10. For the function f graphed in the figure below, find the indicated limit or function value, or state that
it does not exist.
y
3
2
1
-3
(a) lim f (x)
x→−2
(b) f (−2)
(c) lim f (x)
x→−3
(d) f (−3)
(e) lim f (x)
x→1
-2
-1
1
2
3
x
(f) f (1)
(g) lim f (x)
x→2
(h) f (2)
(i) lim+ f (x)
x→1
(j)
11.
lim f (x)
x→−3−
Sketch, as best you can, the graph of a function f that satisfies all of the following properties:
(a) Its domain is the interval [−2, 1].
(b) f (−2) = f (−1) = f (0) = f (1) = f (2) = f (3) = 0.
(c) lim f (x) = 2.
x→−1
(d) lim f (x) = 1.
x→0
(e)
lim f (x) = 0.
x→−2+
(f) lim− f (x) = −1.
x→1
12.
For each of the following limits, sketch a possible graph for the function f and give the appropriate
ǫ-δ definition.
(a) lim f (x) = −3.
x→2
(b) lim f (x) = L.
x→a
(c) lim f (x) = 2,
x→2+
lim f (x) = 6.
x→2−
(d) lim+ f (x) = M .
x→c
13. Given the function f (x) graphed below, give the intervals on which f (x) is continuous.
y
3
2
1
-3
-2
-1
1
2
3
x
14. Let g(x) = x3 − 2x2 + 7x − 3. Is there a c such that 0 ≤ c ≤ 1 and g(c) = 0? Why or why not?