153. The linearization of 2ˣ
a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

1. a. arctan 1

In Exercises 1–4, solve for t.

1. a. e^(-0.3t) = 27

75. a. Find the open intervals on which the function is increasing and decreasing.

g(x) = x(ln x)²

145. The linearization of eˣ at x = 0

a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.

Find the volumes of the solids in Exercises 135 and 136.

135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are

a. circles whose diameters stretch from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).

[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.

139. a. y=arctan(tan x)