Find the tangent line approximation T to the graph of f at the given point. Then complete the table. (Round your answers to four decimal places.) f(x) = sin x, (4, sin 4)

Question

Find the values for f(x) and T(x). The x values are: 3.9, 3.99, 4, 4.01, 4.1

Answer 1

f(x) = sin(x)

f'(x) = cos(x)

The slope of the tangent line at the point where x = 4 is cos(4). The point of tangency is (4, sin(4)).

The equation of the tangent line using point-slope:

y - sin(4) = cos(4)(x - 4)

T(x) = cos(4)(x - 4) + sin(4)

The following Desmos graph has a table showing the function values for f(x) and T(x) at the points where x is very close to or equal to 4.

desmos.com/calculator/uedrncea1m

Note that the function values for T(x) are below the function values for f(x). The is because the curve is concave upward in the vicinity of x = 4 (4th quadrant, so the 2nd derivative -sin(x) is positive), so the tangent line at the point where x = 4 is below the curve.