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.
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Here are the solutions to questions 3 and 4. 3. DIFFERENTIATION Differentiate y = 5 - 4x - 3x^2 from first principles. Step 1: Define f(x) and f(x+h). Given f(x) = 5 - 4x - 3x^2. f(x+h) = 5 - 4(x+h) - 3(x+h)^2 f(x+h) = 5 - 4x - 4h - 3(x^2 + 2xh + h^2) f(x+h) = 5 - 4x - 4h - 3x^2 - 6xh - 3h^2 Step 2: Find f(x+h) - f(x). f(x+h) - f(x) = (5 - 4x - 4h - 3x^2 - 6xh - 3h^2) - (5 - 4x - 3x^2) f(x+h) - f(x) = 5 - 4x - 4h - 3x^2 - 6xh - 3h^2 - 5 + 4x + 3x^2 f(x+h) - f(x) = -4h - 6xh - 3h^2 Step 3: Divide by h. (f(x+h) - f(x))/(h) = (-4h - 6xh - 3h^2)/(h) = (h(-4 - 6x - 3h))/(h) = -4 - 6x - 3h Step 4: Take the limit as h 0. (dy)/(dx) = _h 0 (-4 - 6x - 3h) (dy)/(dx) = -4 - 6x - 3(0) (dy)/(dx) = -4 - 6x The derivative is -4 - 6x. 4. EXPONENTIAL EQUATION If 2^3x+1 - 3(2^2x) + 2^x+1 = 2^x, find the values of x, where x R. Step 1: Rewrite the terms using exponent rules a^m+n = a^m a^n and a^mn = (a^m)^n. 2^3x+1 = 2^3x · 2^1 = (2^x)^3 · 2 2^2x = (2^x)^2 2^x+1 = 2^x · 2^1 = 2 · 2^x Substitute these into the equation: 2(2^x)^3 - 3(2^x)^2 + 2(2^x) = 2^x Step 2: Let u = 2^x. Since x R, u must be positive (u > 0). The equation becomes a cubic polynomial in u: 2u^3 - 3u^2 + 2u = u Step 3: Rearrange the equation and factor. 2u^3 - 3u^2 + 2u - u = 0 2u^3 - 3u^2 + u = 0 Factor out u: u(2u^2 - 3u + 1) = 0 Step 4: Solve for u. This gives two possibilities: 1. u = 0 2. 2u^2 - 3u + 1 = 0 For u=0: Since u = 2^x, 2^x = 0 has no solution because 2^x is always positive. For 2u^2 - 3u + 1 = 0: Factor the quadratic equation. (2u - 1)(u - 1) = 0 This gives two solutions for u: 2u - 1 = 0 2u = 1 u = (1)/(2) u - 1 = 0 u = 1 Step 5: Substitute back u = 2^x to find the values of x. Case 1: u = (1)/(2) 2^x = (1)/(2) 2^x = 2^-1 x = -1 Case 2: u = 1 2^x = 1 2^x = 2^0 x = 0 The values of x are -1, 0. Last free one today — make it count tomorrow, or type /upgrade for unlimited.