A curve passes through (1,−1) if its gradient function is 3×2 + 7 find the equation.

Question

ScanSolve AI Tutor · Accepted 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.

Tafuta equation ya curve kwa kuintegrate gradient function.
Gradient function imetolewa kama $\frac{dy}{dx} = 3x^2 + 2$ .
Ili kupata equation ya curve, tunaintegrate gradient function:
$y = \int (3x^2 + 2)\,dx$
Fanya integration.
Tunatumia sheria ya integration $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ .
$y = 3\left(\frac{x^{2+1}}{2+1}\right) + 2x + C$ $y = 3\left(\frac{x^3}{3}\right) + 2x + C$ $y = x^3 + 2x + C$
Hapa $C$ ni constant of integration.
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:
$C = -1 - 3$ $C = -4$
Andika equation kamili ya curve.
Sasa tunabadilisha thamani ya $C = -4$ kwenye equation kutoka Step 2:
$y = x^3 + 2x - 4$

Point	x	y
y-intercept	0	−4
x-intercept	1.18	0

A curve passes through (1,-1) if its gradient function is 3x2 + 7 find the equation.