a curve is such that dy/dx=4x+7. the line y=2x meets the curve at point 'P'. Given that the gradient of the curve at P is 5. State the coordinates of P.
y=2x^2+7x+c
5=4x+7, so x = -1/2
y(-1/2) = -3+c
at x = -1/2, y=2x = -1, so
-1 = -3+c
c=2
y = 2x^2+7x+2
P=(-1/2,-1)
To find the coordinates of point P where the line y = 2x intersects the curve, we need to consider two things: the equation of the curve and the gradient at point P.
Given that the curve has a derivative dy/dx = 4x + 7, we can find the equation of the curve by integrating this expression with respect to x.
Integrating 4x + 7 with respect to x, we get:
∫(4x + 7) dx = 2x^2 + 7x + C
Where C is the constant of integration.
Now, since the line y = 2x intersects the curve, the coordinates of point P will satisfy both the equation of the curve and the equation of the line.
Setting y in the equation of the curve equal to y in the equation of the line, we have:
2x = 2x^2 + 7x + C
To find C, we can use the information that the gradient at P is 5. The gradient is given by the derivative, so setting the derivative equal to 5, we have:
5 = 4x + 7
Solving this equation for x, we find:
4x = -2
x = -2/4
x = -1/2
Substituting this x-value into the equation of the curve, we can solve for C:
2(-1/2) = 2(-1/2)^2 + 7(-1/2) + C
-1 = 1/2 - 7/2 + C
-1 = -6/2 + C
-1 = -3 + C
C = -1 + 3
C = 2
Therefore, the value of C is 2.
Now we can find the coordinates of point P. Substituting x = -1/2 into the equation of the curve, we get:
y = 2(-1/2)^2 + 7(-1/2) + 2
y = 2(1/4) - 7/2 + 2
y = 1/2 - 7/2 + 2
y = -6/2 + 2
y = -3 + 2
y = -1
Hence, the coordinates of point P are (-1/2, -1).
To find the coordinates of point P where the line y = 2x intersects the curve, we need to equate the gradients of the line and the curve at point P.
First, we are given that the derivative of the curve is dy/dx = 4x + 7.
To find the gradient of the line y = 2x, we can compare it with the standard form of a linear equation, y = mx + c, where m represents the gradient. In this case, m = 2, the slope of the line. So, the gradient of the line is also 2.
To equate the gradients, we have:
2 = 4x + 7 (equating the gradients at point P)
Now, we can solve this equation for the value of x:
2 - 7 = 4x
-5 = 4x
x = -5/4
Thus, the x-coordinate of point P is -5/4.
To find the corresponding y-coordinate, we substitute the value of x back into the equation of the line:
y = 2x
y = 2(-5/4)
y = -10/4
y = -5/2
Therefore, the coordinates of point P where the line y = 2x intersects the curve are (-5/4, -5/2).