Question

G(x)=3sin(x)−1. Use the tangent line to g(x) at x=π/2 to approximate

g(1.5).
a) 0.5
b) 1.5
c) 2.0
d) 2.5

Answer 1

To approximate g(1.5) using the tangent line to g(x) at x = π/2, find the slope of the tangent line, use the point-slope form to find the equation of the tangent line, and substitute x = 1.5 into the equation to find the approximate value of g(1.5).

Step-by-step explanation:

To approximate g(1.5) using the tangent line to g(x) at x = π/2, we first find the slope of the tangent line. The slope of the tangent line is equal to the derivative of g(x) evaluated at x = π/2. The derivative of g(x) is g'(x) = 3cos(x), so g'(π/2) = 3cos(π/2) gives us the slope of the tangent line.

Next, we use the point-slope form of a line to find the equation of the tangent line. The equation of the tangent line is given by y - g(π/2) = g'(π/2)(x - π/2). Plugging in the values, we get y - (3sin(π/2) - 1) = 3cos(π/2)(x - π/2).

To approximate g(1.5), we substitute x = 1.5 into the equation of the tangent line and solve for y. This will give us the approximate value of g(1.5).