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
Hey Jordan, good to see you again. Let's solve Question 1 and Question 2. Question 1: Differentiate the following equations a) y = ( x)/(x^2) Step 1: Apply the quotient rule (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2). Let u = x u' = (1)/(x). Let v = x^2 v' = 2x. (dy)/(dx) = ((1)/(x))(x^2) - ( x)(2x)(x^2)^2 Step 2: Simplify the expression. (dy)/(dx) = (x - 2x x)/(x^4) (dy)/(dx) = (x(1 - 2 x))/(x^4) (dy)/(dx) = (1 - 2 x)/(x^3) b) y = (x^3 + 3x)^5 Step 1: Apply the chain rule (d)/(dx)(f(g(x))) = f'(g(x))g'(x). Let g(x) = x^3 + 3x, so g'(x) = 3x^2 + 3. Let f(u) = u^5, so f'(u) = 5u^4. (dy)/(dx) = 5(x^3 + 3x)^4 · (3x^2 + 3) Step 2: Factor out common terms. (dy)/(dx) = 5(x^3 + 3x)^4 · 3(x^2 + 1) (dy)/(dx) = 15(x^3 + 3x)^4 (x^2 + 1) c) y = e^2x x Step 1: Apply the product rule (d)/(dx)(uv) = u'v + uv'. Let u = e^2x u' = 2e^2x. Let v = x v' = x. (dy)/(dx) = (2e^2x)( x) + (e^2x)( x) Step 2: Factor out the common term e^2x. (dy)/(dx) = e^2x(2 x + x) d) y = 2x^4 - 5x^2 + 6x + 1 Step 1: Differentiate each term using the power rule. (dy)/(dx) = (d)/(dx)(2x^4) - (d)/(dx)(5x^2) + (d)/(dx)(6x) + (d)/(dx)(1) (dy)/(dx) = 2(4x^3) - 5(2x^1) + 6(1) + 0 (dy)/(dx) = 8x^3 - 10x + 6 Question 2: Two sequences U_n = 2n+1 and V_n = U_n^2 - 3 for n 1. I. Write down the first four terms of the sequence (U_n). Step 1: Substitute n=1, 2, 3, 4 into U_n = 2n+1. For n=1: U_1 = 2(1) + 1 = 3 For n=2: U_2 = 2(2) + 1 = 5 For n=3: U_3 = 2(3) + 1 = 7 For n=4: U_4 = 2(4) + 1 = 9 The first four terms are 3, 5, 7, 9. II. Hence, write down the first four terms of the sequence (V_n). Step 1: Substitute the terms of U_n into V_n = U_n^2 - 3. For n=1: V_1 = U_1^2 - 3 = 3^2 - 3 = 9 - 3 = 6 For n=2: V_2 = U_2^2 - 3 = 5^2 - 3 = 25 - 3 = 22 For n=3: V_3 = U_3^2 - 3 = 7^2 - 3 = 49 - 3 = 46 For n=4: V_4 = U_4^2 - 3 = 9^2 - 3 = 81 - 3 = 78 The first four terms are 6, 22, 46, 78. III. Show that V_n = 4n^2 + 4n - 2. Step 1: Substitute U_n = 2n+1 into the expression for V_n. V_n = (2n+1)^2 - 3 Step 2: Expand the squared term and simplify. V_n = (4n^2 + 4n + 1) - 3 V_n = 4n^2 + 4n - 2 IV. Determine whether the sequence U_n and V_n is a GP or AP. Step 1: Check U_n for AP and GP. The terms of U_n are 3, 5, 7, 9, Common difference: U_2 - U_1 = 5 - 3 = 2. U_3 - U_2 = 7 - 5 = 2. Since there is a constant common difference, U_n is an Arithmetic Progression (AP). Common ratio: (U_2)/(U_1) = (5)/(3). (U_3)/(U_2) = (7)/(5). Since the ratios are not equal, U_n is not a GP. Step 2: Check V_n for AP and GP. The terms of V_n are 6, 22, 46, 78, Common difference: V_2 - V_1 = 22 - 6 = 16. $V_3