Question

CAN SOMEONE HELP WITH THIS QUESTION?✨

Answer 1

f'(-3.5) ≈ 0.8292

f'(2) ≈ 5.277

f'(4) ≈ 10.34

Explanation:

You want the approximate derivative of f(x) = 8·1.4^x using h=0.001 at x = {-3.5, 2, 4}.

Derivative

The derivative is approximated by the formula ...

`f'(x)\approx(f(x+h)-f(x))/(h)\\\\f'(x)\approx(f(x+0.001)-f(x))/(0.001)\qquad\text{for $h=0.001$}\\\\f'(x)\approx(8\cdot1.4^(x+0.001)-8\cdot1.4^x)/(0.001)\qquad\text{using the given $f(x)$}`

The calculation for different values of x is tedious, but not difficult.

For example, ...

f'(-3.5) = (8·1.4^-3.499 -8·1.4^-3.5)/0.001 = (2.4648358 -2.4640066)/0.001

= 0.0008292/0.001

f'(-3.5) = 0.8292

The remaining f'(x) values are shown in the attached table in the column f₁(x₂).

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Additional comment

When function evaluation is repeated for different values, it is usually convenient to let a calculator or spreadsheet do the math. You can enter the formula once and have it evaluated as many times as you need.

The formula shown can be simplified to f'(x) ≈ 8000(1.4^(x+.001) -1.4^x), reducing the tedium by a small amount. The second attachment shows a different calculator using this formula.