1
Linear Approximations and Differentials- HW Problems
3
1. Use the linear approximation of 𝑓(𝑥 ) = √𝑥 at 𝑎 = 8 to
3 3
approximate √8.1 and √7.9.
2. Show that the linear approximation of 𝑓(𝑥 ) = tan(𝑥 ) at 𝑎 = 0 is
𝐿(𝑥 ) = 𝑥.
3. Use linear approximations to approximate:
𝑎. (2.001)4
2
𝑏. (8.04) 3
4. Find the differential 𝑑𝑦 of each of the following functions.
𝑎. 𝑦 = sec(√𝑥)
𝑥
𝑏. 𝑦 =
1+𝑥
5. The radius of a circular disk is measured as 6 in. with a maximum
error of 0.3 in.
a. Estimate the maximum error in the calculated area of the disk.
b. Estimate the maximum error in the calculated circumference of the
disk.
c. What is the maximum percentage error in the area of the disk?
Linear Approximations and Differentials- HW Problems
3
1. Use the linear approximation of 𝑓(𝑥 ) = √𝑥 at 𝑎 = 8 to
3 3
approximate √8.1 and √7.9.
2. Show that the linear approximation of 𝑓(𝑥 ) = tan(𝑥 ) at 𝑎 = 0 is
𝐿(𝑥 ) = 𝑥.
3. Use linear approximations to approximate:
𝑎. (2.001)4
2
𝑏. (8.04) 3
4. Find the differential 𝑑𝑦 of each of the following functions.
𝑎. 𝑦 = sec(√𝑥)
𝑥
𝑏. 𝑦 =
1+𝑥
5. The radius of a circular disk is measured as 6 in. with a maximum
error of 0.3 in.
a. Estimate the maximum error in the calculated area of the disk.
b. Estimate the maximum error in the calculated circumference of the
disk.
c. What is the maximum percentage error in the area of the disk?
