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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Welcome back Mary — missed you this week. Here are the solutions to your questions: 1. Differentiate between Differentiation and Integration. Differentiation is the process of finding the rate of change of a function, which gives the slope of the tangent line to the function's graph at any point. Integration is the process of finding the area under a curve, which is essentially the reverse operation of differentiation. 2. Determine the slope of A(-2, -4) and B(-3, -2). Step 1: Use the slope formula m = (y_2 - y_1)/(x_2 - x_1). Given points A(x_1, y_1) = (-2, -4) and B(x_2, y_2) = (-3, -2). Step 2: Substitute the coordinates into the formula. m = (-2 - (-4))/(-3 - (-2)) m = (-2 + 4)/(-3 + 2) m = (2)/(-1) m = -2 The slope is -2. 3. If y = 2x^2 + 5 for 0 x 3.0, find y when x = 0.5. Step 1: Substitute x = 0.5 into the equation. y = 2(0.5)^2 + 5 Step 2: Calculate the value of y. y = 2(0.25) + 5 y = 0.5 + 5 y = 5.5 The value of y is 5.5. 4. If y = ax^n, (dy)/(dx) = \_\_\_\_\_ Using the power rule for differentiation, (d)/(dx)(cx^n) = cnx^n-1. (dy)/(dx) = anx^n-1 5. If y = x^2, (dy)/(dx) = \_\_\_\_\_ Using the power rule, (d)/(dx)(x^n) = nx^n-1. (dy)/(dx) = 2x^2-1 (dy)/(dx) = 2x 6. If y = 2x^5, (dy)/(dx) = \_\_\_\_\_ Using the power rule, (d)/(dx)(cx^n) = cnx^n-1. (dy)/(dx) = 2 · 5x^5-1 (dy)/(dx) = 10x^4 7. A polynomial is \_\_\_\_\_ This question is incomplete. A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. 8. If y = 5x^3 + 4x^2 - 3x + 9, (dy)/(dx) = \_\_\_\_\_ Step 1: Differentiate each term using the power rule. (dy)/(dx) = (d)/(dx)(5x^3) + (d)/(dx)(4x^2) - (d)/(dx)(3x) + (d)/(dx)(9) Step 2: Apply the power rule to each term. (dy)/(dx) = 5(3)x^3-1 + 4(2)x^2-1 - 3(1)x^1-1 + 0 (dy)/(dx) = 15x^2 + 8x - 3 The derivative is 15x^2 + 8x - 3. 9. Find the absolute value of (dy)/(dx) for y = 3x^4 - 7x^3 + 4x^2 - 3x - 4 at x = 2. Step 1: Find the first derivative (dy)/(dx). (dy)/(dx) = (d)/(dx)(3x^4 - 7x^3 + 4x^2 - 3x - 4) (dy)/(dx) = 3(4)x^3 - 7(3)x^2 + 4(2)x^1 - 3(1)x^0 - 0 (dy)/(dx) = 12x^3 - 21x^2 + 8x - 3 Step 2: Substitute x = 2 into the derivative. (dy)/(dx)|_x=2 = 12(2)^3 - 21(2)^2 + 8(2) - 3 (dy)/(dx)|_x=2 = 12(8) - 21(4) + 16 - 3 (dy)/(dx)|_x=2 = 96 - 84 + 16 - 3 (dy)/(dx)|_x=2 = 12 + 16 - 3 (dy)/(dx)|_x=2 = 28 - 3 (dy)/(dx)|_x=2 = 25 Step 3: Find the absolute value. |(dy)/(dx)|_x=2| = |25| = 25 The absolute value of the derivative at x=2 is 25. 10. Find (d^2y)/(dx^2) for y = 3x^4 - 7x^3 + 4x^2 - 3x - 4 at x = 2. Step 1: From Question 9, the first derivative is (dy)/(dx) = 12x^3 - 21x^2 + 8x - 3. Step 2: Find the second derivative (d^2y)/(dx^2) by differentiating (dy)/(dx). (d^2y)/(dx^2) = (d)/(dx)(12x^3 - 21x^2 + 8x - 3) (d^2y)/(dx^2) = 12(3)x^2 - 21(2)x^1 + 8(1)x^0 - 0 (d^2y)/(dx^2) = 36x^2 - 42x + 8 Step 3: Substitute x = 2 into the second derivative. (d^2y)/(dx^2)|_x=2 = 36(2)^2 - 42(2) + 8 (d^2y)/(dx^2)|_x=2 = 36(4) - 84 + 8 (d^2y)/(dx^2)|_x=2 = 144 - 84 + 8 (d^2y)/(dx^2)|_x=2 = 60 + 8 (d^2y)/(dx^2)|_x=2 = 68 The second derivative at x=2 is 68. 11. What is the value of (d^2y)/(dx^2) if y = 3x^4 - 7x^3 + 4x^2 - 3x - 4 at x = -2. Step 1: From Question 10, the second derivative is (d^2y)/(dx^2) = 36x^2 - 42x + 8. Step 2: Substitute x = -2 into the second derivative. (d^2y)/(dx^2)|_x=-2 = 36(-2)^2 - 42(-2) + 8 (d^2y)/(dx^2)|_x=-2 = 36(4) + 84 + 8 (d^2y)/(dx^2)|_x=-2 = 144 + 84 + 8 (d^2y)/(dx^2)|_x=-2 = 228 + 8 (d^2y)/(dx^2)|_x=-2 = 236 The value of the second derivative at x=-2 is 236. 12. If y = x, (dy)/(dx) = \_\_\_\_\_ The derivative of x is x. (dy)/(dx) = x 13. If y = x, (dy)/(dx) = \_\_\_\_\_ The derivative of x is - x. (dy)/(dx) = - x 14. What is a Curve? A curve is a continuous and smooth line that is not necessarily straight. In mathematics, it often represents the graph of a function or a set of points satisfying a certain equation. 15. If y = ( x)/( x), (dy)/(dx) = \_\_\_\_\_ Step 1: Recognize that y = ( x)/( x) = x. Step 2: Differentiate y = x. (dy)/(dx) = (d)/(dx)( x) (dy)/(dx) = ^2 x Alternatively, using the quotient rule: Let u = x and v = x. Then (du)/(dx) = x and (dv)/(dx) = - x. The quotient rule is (d)/(dx)((u)/(v)) = (vdu)/(dx) - u(dv)/(dx)v^2. (dy)/(dx) = ( x ( x) - x (- x))/(( x)^2) (dy)/(dx) = (^2 x + ^2 x)/(^2 x) Using the identity ^2 x + ^2 x = 1: (dy)/(dx) = (1)/(^2 x) (dy)/(dx) = ^2 x The derivative is ^2 x. Send me the next one 📸