AP Calculus
More Limit Definition
and Interpretation of
Derivative
HW 2.6
Name:
Date:
Collaborator:
1.) Write the limit definition of derivative:
Answer each question if possible. If not, explain what the question is asking you to find, stating
the specific function.
1.) lim
ℎ→0
(𝑥+ℎ)2 −𝑥 2
ℎ
√𝑥+ℎ−√𝑥
ℎ
ℎ→0
3.) lim
2.) lim
(𝑥+ℎ)2 −3𝑥−3ℎ−(𝑥 2 −3𝑥)
ℎ
ℎ→0
4.) lim
cos2(𝑥+ℎ)−cos2 𝑥
ℎ→0
ℎ
Answer each question if possible. If not, explain what the question is asking you to find, stating
the specific function and x-location.
5.) lim
ℎ→0
8.) lim
ℎ→0
(1+ℎ)3 −13
ℎ
4
−4
3
√𝑥+ℎ
ℎ
6.) lim
ℎ→0
9.) lim
ℎ→0
ln(𝑒+ℎ)−1
ℎ
7.) lim
3
ℎ→0
2(2+ℎ)4 −2∗16
ℎ
10.) lim
ℎ→0
√27+ℎ−3
ℎ
2 sin2 (𝜋+ℎ)−0
ℎ
1
AP Calculus
More Limit Definition
and Interpretation of
Derivative
HW 2.6
Name:
Date:
Collaborator:
11.) Use the function below to answer the questions that follow:
y




x

   
  














a.) lim 𝑓(𝑥)
b.) lim
𝑥→3
d.) lim
ℎ→0
𝑓(3+ℎ)−𝑓(3)
ℎ→0
𝑓(−2+ℎ)−𝑓(−2)
ℎ
e.) lim 𝑓(𝑥)
𝑥→4
ℎ
c.) 𝑓(3)
f) lim 𝑓(𝑥)
𝑥→−∞
g.) Is the function continuous at 𝑥 = −4? Justify your answer.
2


AP Calculus
More Limit Definition
and Interpretation of
Derivative
HW 2.6
Name:
Date:
Collaborator:
12.) Let the smoking rate among high-school seniors at time 𝑡 be 𝑆(𝑡). The table (from the
Institute of Social Research, University of Michigan) gives the percentage of seniors who
reported that they had smoked one or more cigarettes per day during the past 30 days.
a.) What is the meaning of 𝑆′(𝑡)? What are its units?
b.) Approximate 𝑆′(1983) and interpret its meaning.
13.) A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited
entries in a special box between noon (𝑡 = 0) and 8 PM (𝑡 = 8). The number of entries in the
box 𝑡 hours after noon is modeled by a differentiable function 𝐸 for 0 ≤ 𝑡 ≤ 8. Values of 𝐸(𝑡),
in hundreds of entries, at various times 𝑡 are shown in the table above.
𝑡 (ℎ𝑜𝑢𝑟𝑠)
0
2
5
7
8
𝐸(𝑡) (ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑠 𝑜𝑓 𝑒𝑛𝑡𝑟𝑖𝑒𝑠)
0
4
13
21
23
a.) Use the data in the table to approximate the rate at which entries were being deposited at time
𝑡 = 6. Interpret the meaning of your answer.
3
AP Calculus
More Limit Definition
and Interpretation of
Derivative
HW 2.6
Name:
Date:
Collaborator:
14.) The number 𝑁 (in thousands) of cellular phone subscribers in Malaysia is a function of time
in years given by 𝑁(𝑡).
a.) Interpret the meaning of 𝑁′(𝑡)
b.) Interpret the meaning of 𝑁 ′ (2) = 500.
15.) Find the equation of the line tangent to the function 𝑦 = −3𝑥 2 − 4𝑥 + 2 𝑎𝑡 𝑥 = −1
16.) Find the equation of the line normal to the function 𝑦 = −𝑥 3 + 2𝑥 − 1 𝑎𝑡 𝑥 = 0
17.) Find the equation of the line tangent to the function 𝑦 = −2𝑥 2 + 5𝑥 − 2 𝑎𝑡 𝑥 = 2
4