   Chapter 2.R, Problem 55E

Chapter
Section
Textbook Problem

# Find a parabola y = a x 2 + b x + c that passes through the point (1, 4) and whose tangent lines at x = − 1 and x = 5 have slopes 6 and -2, respectively.

To determine

To find:

an equation of parabola from given information.

Explanation

1) Concept

Slope of a curve at a given point is nothing but the derivative of the function at that point.So here equate the derivative to the slope, this step gives equation in terms of a and b, solve these equation simultaneously and find values of a and b.

Plug the values of a and b, ordered pair ( 1, 4 ) in equation y=ax2+bx+c and solve for c.

2) Given:

Curve y=ax2+bx+c is passing through (1,4), have two tangent lines at  x=-1 and x=5, whose slope is 6 and -2 respectively.

3) Calculations:

Find the derivative of y with respect to x, using standard differential rules.

y'=2ax+b

Derivative of a function at a point gives slope of tangent to the curve at that point, so from given information:

y'=6 when x=-1 So plug these values in equation of  y'.

That is:

6=2a-1+b

6=-2a+b        ---------(1)

y'=-2  when x=5 So plug these values in equation of  y'.

That is:     -2=2a5+b

-2=10a+b   ----------(2)

Solve equation for b, it become:

b=-10a-2  ----------(3)

Plug the expression of b in equation (1), it becomes:

6=-2a+-10a-2

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