Module 16 - The Fundamental Theorem

Module 16 - Answers

Lesson 1

Answer 1

16.1.1 By the pattern shown, the area under the curve between x = 0 and x = 5 should be

, which is confirmed by the TI-89.

Answer 2

16.1.2

The area under the curve between x = 0 and x = b is given by .

Answer 3

16.1.3

Area Function

Curve Function f(x) = x² f(x) = x³ f(x) = x⁴

Answer 4

16.1.4 The derivative of the area function is the curve function.

Lesson 2

Answer 1

16.2.1 Evaluate

for each value of x by using the integral key and entering

x 1.0 1.5 2.0 2.5 3.0 3.5 4.0

area 0 0.405 0.693 0.916 1.099 1.253 1.386

Answer 2

16.2.2

The regression equation is approximately y = ln x

Answer 3

16.2.3

= ln x

Answer 4

16.2.4 The derivative of the area function, A(x) = ln x, is

, the curve function.

Answer 5

16.2.5 The height of the curve is increasing, so the slopes of the area function are increasing and its graph is concave upward. An approximate area function graph is shown below. The actual values of the area function are not as important as the graph's shape.