Question

Find f(a), f(ah), and the difference quotient for the function given below, where h ≠ 0.

f(x) = 2x⁷

a) f(a) = ?
b) f(ah) = ?
c) The difference quotient (f(ah) - f(a))/h = ?

Answer 1

To find f(a) for the function f(x) = 2x⁷, you substitute a into the function and get 2a⁷. For f(ah), you substitute ah and simplify to get 2a⁷h⁷. The difference quotient, (f(ah) - f(a))/h, simplifies to 2a⁷(h⁶ + h⁵ + h⁴ + h³ + h² + h + 1).

Step-by-step explanation:

To find f(a), f(ah), and the difference quotient for the function f(x) = 2x⁷, we will evaluate the function at the given points and then calculate the difference quotient.

1. To find f(a), simply substitute a into the function:

   f(a) = 2a⁷

2. To find f(ah), we substitute ah into the function:

   f(ah) = 2(ah)⁷ = 2a⁷h⁷

3. The difference quotient is calculated as follows:

   (f(ah) - f(a)) / h = (2a⁷h⁷ - 2a⁷) / h

   Factor out 2a⁷ from both terms in the numerator:

   = 2a⁷(h⁷ - 1) / h

   We can see that h is a common factor in all the terms of h⁷, except for one, which allows us to simplify:

   = 2a⁷(h⁶ + h⁵ + h⁴ + h³ + h² + h + 1)