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: Differentiate y = 3x^6. Using the power rule for differentiation, (d)/(dx)(ax^n) = anx^n-1: (dy)/(dx) = 3 × 6 x^6-1 (dy)/(dx) = 18x^5 The derivative is 18x^5. Step 2: Find (dy)/(dx) of y = 2x^3 - 3x^2 + 12x. Differentiate each term using the power rule: (dy)/(dx) = (d)/(dx)(2x^3) - (d)/(dx)(3x^2) + (d)/(dx)(12x) (dy)/(dx) = (2 × 3 x^3-1) - (3 × 2 x^2-1) + (12 × 1 x^1-1) (dy)/(dx) = 6x^2 - 6x + 12x^0 (dy)/(dx) = 6x^2 - 6x + 12 The derivative is 6x^2 - 6x + 12. Step 3: Integrate (10x^3 + 5x + 3x)\,dx. First, simplify the integrand: (10x^3 + 8x)\,dx Using the power rule for integration, ax^n\,dx = ax^n+1n+1 + C: 10x^3\,dx + 8x\,dx 10x^3+13+1 + 8x^1+11+1 + C 10(x^4)/(4) + 8(x^2)/(2) + C (5)/(2)x^4 + 4x^2 + C The integral is (5)/(2)x^4 + 4x^2 + C. Step 4: Construct the following angles. These steps describe how to construct the angles using a compass and straightedge. i) 90^ • Draw a line segment and mark a point P on it. • With P as the center, draw an arc intersecting the line at two points, A and B. • With A and B as centers, and a radius greater than AP, draw two arcs intersecting each other at point C. • Join P and C. The angle CPA (or CPB) is 90^. ii) 45^ • Construct a 90^ angle, say CPB, as described above. • Bisect the 90^ angle. With B and C as centers, and a radius greater than half of BC, draw two arcs intersecting each other at point D. • Join P and D. The angle DPB is 45^. iii) 60^ • Draw a line segment and mark a point P on it. • With P as the center, draw an arc intersecting the line at point A. • With A as the center and the same radius, draw another arc intersecting the first arc at point B. • Join P and B. The angle BPA is 60^. iv) 120^ • Draw a line segment and mark a point P on it. • With P as the center, draw an arc intersecting the line at point A. • With A as the center and the same radius, draw an arc intersecting the first arc at point B (this forms 60^). • With B as the center and the same radius, draw another arc intersecting the first arc at point C. • Join P and C. The angle CPA is 120^. v) 150^ • Draw a line segment AB and mark a point P on it. • With P as the center, draw an arc intersecting AB at points X and Y. • With X as the center and the same radius, draw an arc intersecting the first arc at Z. This forms ZPX = 60^. • Bisect ZPX. With X and Z as centers, draw arcs intersecting at M. Join P and M. This forms MPX = 30^. • The angle MPY is 180^ - 30^ = 150^.