Calculus 4.
6 Approximating with Local Linearity Notes
Write your questions
and thoughts here!
The tangent line of the function 𝑓𝑓(𝑥𝑥) at 𝑥𝑥 = 𝑎𝑎 can give you an approximate value of 𝑓𝑓(𝑥𝑥) for
points close to 𝑥𝑥 = 𝑎𝑎.
Concave UP with a Tangent Line Concave DOWN with a Tangent Line
UNDERESTIMATE OVERESTIMATE
1. 𝑓𝑓 is concave up on its domain with 𝑓𝑓(4) = 5 and 𝑓𝑓 ′ (4) = 3.
a. What is the estimate for 𝑓𝑓(3.8) using the local linear approximation for 𝑓𝑓 at 𝑥𝑥 = 4?
b. Is it an underestimate or overestimate? Explain.
2. The function 𝑓𝑓(𝑥𝑥) = 5𝑥𝑥 − 2𝑥𝑥 3 − 2 is concave down at 𝑥𝑥 = 1?.
a. Find the tangent line of 𝑓𝑓 at 𝑥𝑥 = 1.
b. What is the estimate for 𝑓𝑓(1.1) using the local linear approximation for 𝑓𝑓 at 𝑥𝑥 = 1?
c. Is it an underestimate or overestimate? Explain.
𝑑𝑑𝑑𝑑
3. Consider the differential equation 𝑑𝑑𝑑𝑑 = 𝑒𝑒 𝑑𝑑 (2𝑥𝑥 2 − 5𝑥𝑥). Let 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) be the particular
solution to the differential equation with the initial condition 𝑓𝑓(2) = 0.
a. Write an equation for the line tangent to the graph of 𝑓𝑓 at the point (2,0).
b. Use the tangent line to approximate 𝑓𝑓(2.2).
4.6 Approximating with Local Linearity
Calculus
Practice
For each differential equation, let 𝒚𝒚 = 𝒇𝒇(𝒙𝒙) be the particular solution to the differential equation with the
given initial condition.
𝑑𝑑𝑑𝑑 𝜋𝜋 𝑑𝑑𝑑𝑑 4𝑑𝑑
1. = (5 − 𝑦𝑦) sin 𝑥𝑥 and 𝑓𝑓 � � = 2. 2. 𝑑𝑑𝑑𝑑 = − 𝑑𝑑 and 𝑓𝑓(1) = 3.
𝑑𝑑𝑑𝑑 2
a. Write an equation for the line tangent to the a. Write an equation for the line tangent to the
𝜋𝜋 graph of 𝑓𝑓 at the point (1,3).
graph of 𝑓𝑓 at the point � , 2�.
2
b. Use the tangent line to approximate 𝑓𝑓(1.1).
b. Use the tangent line to approximate 𝑓𝑓(1.5).
Answer the questions for each function listed.
𝜋𝜋 𝑒𝑒 2𝑥𝑥
3. 𝑓𝑓(𝑥𝑥) = 2 cos 𝑥𝑥 + 1 is concave down on �0, �. 4. 𝑓𝑓(𝑥𝑥) = 𝑑𝑑+1 is concave up on 𝑥𝑥 > −1.
2
a. What is the estimate for 𝑓𝑓(1) using the local a. What is the estimate for 𝑓𝑓(0.1) using the local
𝜋𝜋
linear approximation for 𝑓𝑓 at 𝑥𝑥 = ? Give an linear approximation for 𝑓𝑓 at 𝑥𝑥 = 0?
2
exact answer (no rounding).
b. Is it an underestimate or overestimate? b. Is it an underestimate or overestimate?
Explain. Explain.
5. 𝑓𝑓(𝑥𝑥) = −√4 − 𝑥𝑥 is concave up on its domain. 6. 𝑓𝑓 is concave down and 𝑓𝑓(3) = −1 and 𝑓𝑓 ′ (3) = 2.
a. What is the estimate for 𝑓𝑓(1.9) using the local a. What is the estimate for 𝑓𝑓(3.2) using the local
linear approximation for 𝑓𝑓 at 𝑥𝑥 = 2? Round to linear approximation for 𝑓𝑓 at 𝑥𝑥 = 3?
three decimal places.
b. Is it an underestimate or overestimate?
Explain.
b. Is it an underestimate or overestimate?
Explain.
7. 𝑓𝑓 is concave up and 𝑓𝑓(−5) = 2 and 𝑓𝑓 ′ (−5) = −1. 8. 𝑓𝑓 is concave down and 𝑓𝑓(2) = 1 and 𝑓𝑓 ′ (2) = −3.
a. What is the estimate for 𝑓𝑓(−5.1) using the a. What is the estimate for 𝑓𝑓(1.9) using the local
local linear approximation for 𝑓𝑓 at 𝑥𝑥 = −5? linear approximation for 𝑓𝑓 at 𝑥𝑥 = 2?
b. Is it an underestimate or overestimate? b. Is it an underestimate or overestimate?
Explain. Explain.
4.6 Approximating with Local Linearity Test Prep
9. Let 𝑓𝑓 be the function given by 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 2 − 4𝑥𝑥 + 2. The tangent line to the graph of 𝑓𝑓 at 𝑥𝑥 = 1 is
used to approximate values of 𝑓𝑓(𝑥𝑥). Which of the following is the smallest value of 𝑥𝑥 for which the
error resulting from this tangent line approximation is more than 0.5?
[Hint for your calculator use: Create a table to compare values of two functions.]
(A) 1.3 (B) 1.4 (C) 1.5 (D) 1.6 (E) 1.7
ANSWER: B
10. The depth of snow in a field is given by the twice-differentiable function 𝑆𝑆 for 0 ≤ 𝑡𝑡 ≤ 12, where
𝑆𝑆(𝑡𝑡) is measured in centimeters and time 𝑡𝑡 is measured in hours. Values of 𝑆𝑆 ′ (𝑡𝑡), the derivative of
𝑆𝑆, at selected values of time 𝑡𝑡 are shown in the table above. It is known that the graph of 𝑆𝑆 is
concave down for 0 ≤ 𝑡𝑡 ≤ 12.
𝑡𝑡
0 1 4 9 12
(hours)
𝑆𝑆 ′ (𝑡𝑡)
(centimeters per 1.8 2.4 2.0 1.6 1.3
hour)
a. Use the data in the table to approximate 𝑆𝑆 ′′ (10). Show the computations that lead to your
answer. Using correct units, explain the meaning of 𝑆𝑆 ′′ (10) in the context of the problem.
b. Is there a time 𝑡𝑡, for 0 ≤ 𝑡𝑡 ≤ 12, at which the depth of snow is changing at a rate of 1.5
centimeters per hour? Justify your answer?
c. At time 𝑡𝑡 = 4, the depth of snow is 28 centimeters. Use the line tangent to the graph of 𝑆𝑆 at
𝑡𝑡 = 4 to approximate the depth of the snow at time 𝑡𝑡 = 6. Is the approximation an
underestimate or an overestimate of the actual depth of snow at time 𝑡𝑡 = 6? Justify your answer.
