AP Calculus BC
Unit 1: Problem Set #3
Name ___________________________
No Calculator. Show all work or explain your reasoning and use correct notation.
⎧1
⎪ 2 x + 5, x > 2
⎪
Given f ( x ) = ⎨6,
x=2
⎪ x + 4,
x<2
⎪
⎩
1.
a.
Write f ′ as a piecewise function.
f CxJ
f
f
nim
in.no
µg
z
LQ2
fly
b.
E
Daz
Determine if f is differentiable at x = 2.
Lim
Yost
Iim
f
Koz Exts
6
im
h
n
hIo l
Give a reason for your answer.
tim
x 2
im
XD
Xt4
6
ta
2. a. Use the definition of derivative to find f ′ ( x ) for f ( x ) = 2 x .
f
D info thfCX
2
linmgo2xth
h
2Fhyzg
25
ljngo4Lxth
limo
353 1 3525
51 1
30
6525
1255
n
nxzRD
f
fo
n Io
z
xrtnt
zr
b. Use f ′ ( x ) to determine at what point the tangent line to f has a slope of 2.
4
4Fx
4
3
2574
I
86
iI
ex
c. Write the equation of the tangent line to f at x = 9 .
f'CaJ
ra
Point
G
tG
3. The position, in meters, of a continuously moving object for 1 ≤ t ≤ 1.8 seconds is given in the table
t (sec)
position
1
–2.4
1.2
–1.663
1.4
–0.744
1.6
0.399
1.8
1.825
a. Find average velocity 1 ≤ t ≤ 1.6. Include units on your answer.
C 2.4J
i
2
Y.IT
b.
399J
7
466sIs
Use symmetric difference quotient to estimate instantaneous velocity at t = 1.5 sec. Include units.
C4
744
399to
c.
6
a44
399
l
4z3
5.7l5mD
Is average or instantaneous velocity the slope of a secant line?
average
d.
Find the minimum number of times the object had position 1 meter. Justify your answer.
ftp.isconarobonftn.SI
f
16 L
Lfa
L
By the IVT, there exists a solution on
the interval(0, 5) such as that f(c) = 1
Selected Answers:
1. b. no, explain
8
atleast
2. b. (1/4, 1)
3. a. 4.665 m/s
b. ≈ 5.715 m/s
c. average
d. Once, explain using IVT and data