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
Huu hapa ni ufumbuzi wa swali la 1: Swali linasema gradient function ya curve ni 3x^2 + 2 na curve inapitia point (1, -1). Tunatakiwa kutafuta equation ya curve. Step 1: Tafuta equation ya curve kwa kuintegrate gradient function. Gradient function imetolewa kama (dy)/(dx) = 3x^2 + 2. Ili kupata equation ya curve, tunaintegrate gradient function: y = (3x^2 + 2)\,dx Step 2: Fanya integration. Tunatumia sheria ya integration x^n\,dx = x^n+1n+1 + C. y = 3(x^2+12+1) + 2x + C y = 3((x^3)/(3)) + 2x + C y = x^3 + 2x + C Hapa C ni constant of integration. Step 3: Tumia point (1, -1) kutafuta thamani ya C. Curve inapitia point (1, -1), kwa hiyo tunaweza kubadilisha x=1 na y=-1 kwenye equation tuliyoipata: -1 = (1)^3 + 2(1) + C -1 = 1 + 2 + C -1 = 3 + C Ili kutafuta C, tunatoa 3 pande zote mbili: C = -1 - 3 C = -4 Step 4: Andika equation kamili ya curve. Sasa tunabadilisha thamani ya C = -4 kwenye equation kutoka Step 2: y = x^3 + 2x - 4 Equation ya curve ni: y = x^3 + 2x - 4 Tuma swali linalofuata 📸