Find a parabola of the form given below that has slope m1 at x1, slope m2 at x2, and passes through the point P.
y=ax^2 + bx + c
m1 = 8, x1 = 11
m2 = 7, x2 = 9
P = (5,7)
y = ?
I don't know where to begin. I can't find anywhere in my textbook that teaches how to solve this type of problem.
The slope of the parabola equals to the derivative of y:
y'(x)= dy/dx = d(ax^2 + bx + c)/dx
=2ax+b
Knowing that the slope at x1=11 is 8, we get
y'(11)=8
2a(11)+b=8 .....(1)
Similarly,
y'(9)=7
2a(9)+b=7.......(2)
From (1) and (2), solve for a and b.
c can be found by substituting the coordinates of x and y in P(5,7), i.e.
y(5)=7, or
a(5)²+b(5)+c = 7 ....(3)
Finally, from the values of a,b and c calculated above, substitute into the equations (1), (2) and (3) to make sure that all conditions are satisfied. If not, review the calculations.
Aha! Thank you.
Remember that the slope at a given point is equal to the first derivative at that point.
so if y = ax^2 + bx + c
dy/dx = 2ax + b
when x = 11, dy/dx = 8 ---> 22a + b = 8
when x = 9 , dy/dx = 7 ----> 18a + b = 7
subtract them ---> 4a = 1
a = 1/4
then 18(1/4) + b = 7
b = 5/2
so y = (1/4)x^2 + (5/2)x + c
plug in (5,7)
7 = (1/4)(25) + (5/2)(5) + c
c = -47/5
so finally
y = (1/4)x^2 + (5/2)x - 47/5
check my arithmetic.
To find the equation of the parabola, we can use the given information about the slopes and the point it passes through.
Step 1: Calculate the values of a, b, and c in the equation y = ax^2 + bx + c, using the known values of x1, x2, m1, and m2.
First, we need to find the derivatives of the given equation to obtain expressions for the slopes at x1 and x2:
y' = 2ax + b (derivative of ax^2 with respect to x)
Now, we can substitute the known values of x1, m1, x2, and m2 into the derivative expressions and solve for a and b:
For x1 = 11 and m1 = 8:
8 = 2(11)a + b Equation 1
For x2 = 9 and m2 = 7:
7 = 2(9)a + b Equation 2
Solve Equations 1 and 2 simultaneously to find the values of a and b.
Step 2: Once we have obtained the values of a and b, substitute them into the equation y = ax^2 + bx + c.
Now we will use the given point P(5,7) to solve for c:
Substitute x = 5 and y = 7 into the equation y = ax^2 + bx + c:
7 = a(5^2) + b(5) + c
Solve the equation to find the value of c.
Step 3: Now that we have the values of a, b, and c, substitute them into the equation y = ax^2 + bx + c, and simplify to find the equation of the parabola.
Once you have computed the values of a, b, and c, substitute them into the equation y = ax^2 + bx + c, and simplify to find the equation of the parabola that satisfies the given conditions.