This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Calculate the function values at x = -1 and x = 3. Given f(x) = -2x^3. For x = -1: f(-1) = -2(-1)^3 = -2(-1) = 2 For x = 3: f(3) = -2(3)^3 = -2(27) = -54 Step 2: Apply the formula for the average gradient. The average gradient is given by (f(x_2) - f(x_1))/(x_2 - x_1). Average gradient = (f(3) - f(-1))/(3 - (-1)) = (-54 - 2)/(3 + 1) = (-56)/(4) Average gradient = -14 The average gradient between x = -1 and x = 3 is -14. Step 1: Write down the definition of the derivative from first principles. f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Find f(x+h). Given f(x) = -2x^3. f(x+h) = -2(x+h)^3 Expand (x+h)^3: (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 Substitute this back into f(x+h): f(x+h) = -2(x^3 + 3x^2h + 3xh^2 + h^3) = -2x^3 - 6x^2h - 6xh^2 - 2h^3 Step 3: Calculate f(x+h) - f(x). f(x+h) - f(x) = (-2x^3 - 6x^2h - 6xh^2 - 2h^3) - (-2x^3) f(x+h) - f(x) = -2x^3 - 6x^2h - 6xh^2 - 2h^3 + 2x^3 f(x+h) - f(x) = -6x^2h - 6xh^2 - 2h^3 Step 4: Divide by h and simplify. (f(x+h) - f(x))/(h) = (-6x^2h - 6xh^2 - 2h^3)/(h) Factor out h from the numerator: (h(-6x^2 - 6xh - 2h^2))/(h) = -6x^2 - 6xh - 2h^2 Step 5: Take the limit as h 0. f'(x) = _h 0 (-6x^2 - 6xh - 2h^2) Substitute h=0: f'(x) = -6x^2 - 6x(0) - 2(0)^2 f'(x) = -6x^2 The derivative f'(x) from first principles is -6x^2. Step 1: Use the result from 1.1.2, which is f'(x) = -6x^2. Substitute x = -1 into the derivative function: f'(-1) = -6(-1)^2 f'(-1) = -6(1) f'(-1) = -6 The value of f'(-1) is -6. Send me the next one 📸