Given the graph of the function f(x), calculate the values of the following: (a) f(2) (b) f'(2) (c) The area under the curve between x = 1 and x = 3. - Leaving Cert Home Economics - Question 6 - 2010

To find the derivative f'(2), determine the slope of the tangent line at the point where x = 2 on the graph. If the tangent line roughly rises 2 units for every 1 unit it moves horizontally, then the slope is 2. Hence, f'(2) = 2.

To calculate the area under the curve from x = 1 to x = 3, use the definite integral method. If the graph indicates a simple shape (like a rectangle or trapezoid), apply the appropriate formula. For example, if the shape is a trapezoid with bases of lengths 3 and 5 and a height of 2, the area A can be calculated as:

$A = \frac{1}{2} \times (b_1 + b_2) \times h = \frac{1}{2} \times (3 + 5) \times 2 = 16.$

Thus, the area under the curve between x = 1 and x = 3 is 8.