Calculus 1 Review Problems for Test 1
1)
State the definition of (a) function, (b) limit (ε – δ definition), and compute the largest value of δ that
satisfies the ε – δ condition for the following statement:
lim (5x + 1) = 11 for ε = .25
x →2
2)
Without any calculator, sketch the graphs on the same set of axes:
a) f(x) = x
b) f(x) = x – 2
c)
f(x) = − x + 3
3)
Without a calculator, sketch the graphs of:
a) f(x) = tan x
b)
f(x) = 3 sin x + 2
4)
Compute each of these values without the aid of a calculator:
7
a) convert 24 degrees to radians
b) convert
radians to degrees
6
−2 
3
c) sin
d) tan 12
e) sec
3
4
5)
Solve for x 0≤x 2  :
6)
Prove lim
(1 - 4x) = -7 by using the ε – δ definition of a limit.
x →2
7)
Compute the limits. Show relevant work!
a) lim
(5x2 – 3x + 2)
x →2
b) lim
x →5
9)
g)
h)
2
1
b) cos x− 2 cos x=0
lim
x →4 +
12 − 3 x
lim 12 − 3 x
x →4 −
lim+
5x + 2
2x − 6
lim−
5x + 2
2x − 6
12 − 3 x
c) lim
x →4
i)
−x+7
d) xlim
→7 −
j)
(55)
e) xlim
→10
k)
 sin 6 x 
lim 

x→ 0
 3x 
l)
 3 x + 1 + sin x − cos x 
lim 

x→ 0
x


f) lim
x →9
8)
x 2 + x − 30
10 − 2 x
− 3
2
a) sin(x) =
x−9
x−3
x →3
x →3
 x 2 if 0 < x < 1
Evaluate f(x) and find the left, right,
two-sided limits at x = 0, 1, 2, and 3. f ( x ) =  x if 1 ≤ x < 3
2 x
if x > 3

If f(x) = x + 3 and g(x) = x2 – 1, find
a)
g( 13 )
b)
f(g(x))
c)
g(f(6))
e)
the domain of f(x)
f) the domain of
3
g x





d)
g(x+h)