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
Here are the solutions for questions A, B, and C. A) If f(x) = 3x^2 - 2x + 1, find f'(x) from first principles. Step 1: Write down the definition of the derivative from first principles and find f(x+h). The definition of the derivative is: f'(x) = _h 0 (f(x+h) - f(x))/(h) Given f(x) = 3x^2 - 2x + 1, we find f(x+h): f(x+h) = 3(x+h)^2 - 2(x+h) + 1 f(x+h) = 3(x^2 + 2xh + h^2) - 2x - 2h + 1 f(x+h) = 3x^2 + 6xh + 3h^2 - 2x - 2h + 1 Step 2: Calculate f(x+h) - f(x). f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 - 2x - 2h + 1) - (3x^2 - 2x + 1) f(x+h) - f(x) = 3x^2 + 6xh + 3h^2 - 2x - 2h + 1 - 3x^2 + 2x - 1 f(x+h) - f(x) = 6xh + 3h^2 - 2h Step 3: Divide by h and simplify. (f(x+h) - f(x))/(h) = (6xh + 3h^2 - 2h)/(h) (f(x+h) - f(x))/(h) = (h(6x + 3h - 2))/(h) (f(x+h) - f(x))/(h) = 6x + 3h - 2 Step 4: Take the limit as h 0. f'(x) = _h 0 (6x + 3h - 2) f'(x) = 6x + 3(0) - 2 f'(x) = 6x - 2 B) Differentiate y = (3x^2 - 2x + 1)/(sqrt(x)) with respect to x. Step 1: Rewrite the function by expressing sqrt(x) as x^1/2 and dividing each term in the numerator. y = (3x^2)/(x^1/2) - (2x)/(x^1/2) + (1)/(x^1/2) y = 3x^2 - 1/2 - 2x^1 - 1/2 + x^-1/2 y = 3x^3/2 - 2x^1/2 + x^-1/2 Step 2: Differentiate each term using the power rule, (d)/(dx)(ax^n) = anx^n-1. (dy)/(dx) = (d)/(dx)(3x^3/2) - (d)/(dx)(2x^1/2) + (d)/(dx)(x^-1/2) (dy)/(dx) = 3 ((3)/(2)) x^(3)/(2) - 1 - 2 ((1)/(2)) x^(1)/(2) - 1 + (-(1)/(2)) x^-(1)/(2) - 1 (dy)/(dx) = (9)/(2) x^1/2 - 1 x^-1/2 - (1)/(2) x^-3/2 Step 3: Simplify the expression. (dy)/(dx) = (9)/(2) sqrt(x) - (1)/(sqrt(x)) - (1)/(2xsqrt(x)) C) Find the equation of the tangent to the curve y = x^3 - 2x^2 + 3x - 1 at the point where x=2. Step 1: Find the y-coordinate of the point of tangency. Substitute x=2 into the equation of the curve: y = (2)^3 - 2(2)^2 + 3(2) - 1 y = 8 - 2(4) + 6 - 1 y = 8 - 8 + 6 - 1 y = 5 The point of tangency is (2, 5). Step 2: Find the derivative (dy)/(dx) to determine the slope function. y = x^3 - 2x^2 + 3x - 1 (dy)/(dx) = 3x^2 - 2(2x) + 3(1) - 0 (dy)/(dx) = 3x^2 - 4x + 3 Step 3: Evaluate the derivative at x=2 to find the slope (m) of the tangent. m = 3(2)^2 - 4(2) + 3 m = 3(4) - 8 + 3 m = 12 - 8 + 3 m = 7 Step 4: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), to find the equation of the tangent. Using the point (x_1, y_1) = (2, 5) and slope m=7: y - 5 = 7(x - 2) y - 5 = 7x - 14 y = 7x - 14 + 5 y = 7x - 9 That's 2 down. 3 left today — send the next one.