Question

If f and g are functions of time, and at time t = 3, f equals 9 and is rising at a rate of 5 units per second, and g equals 7 and is rising at a rate of 9 units per second, then the product fg equals__ and is rising at a rate of__units per second.

A) 45; 14
B) 63; 14
C) 63; 77
D) 20; 33

Answer 1

The product of the functions f and g at t = 3 is 63, and using the product rule for differentiation, we determine the rate at which the product is rising to be 116 units per second at that moment.

Step-by-step explanation:

If f and g are functions of time, and at time t = 3, f equals 9 and is rising at a rate of 5 units per second, and g equals 7 and is rising at a rate of 9 units per second, we need to find the product fg and the rate at which it is rising. The product of the functions at t = 3 is given by f(3) × g(3) = 9 × 7 = 63, so the first part of our solution is 63.

To find the rate at which the product is rising, we use the product rule for differentiation, which states (fg)' is f'g + fg'. Plugging in our values, we get (5 × 7) + (9 × 9) = 35 + 81 = 116. Hence, the rate at which the product fg is rising at t = 3 is 116 units per second. Therefore, the correct answer is none of the options provided as they all have the incorrect rate of change.