   Chapter 3.1, Problem 76E

Chapter
Section
Textbook Problem

Suppose the curve y = x4 + ax3 + bx2 + cx + d has a tangent line when x = 0 with equation y = 2x + 1 and a tangent line when x = 1 with equation y = 2 – 3x. Find the values of a, b, c, and d.

To determine

To find: The value of the point a, b, c and d in the given equation of the curve.

Explanation

Given:

The equation of the curve y=x4+ax3+bx2+cx+d. (1)

The equation of tangent lines y=2x+1 and y=23x to the curve at x=0and x=1.

Derivative rules:

(1) Constant multiple rule: ddx(cf)=cddx(f)

(2) Power rule: ddx(xn)=nxn1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

Calculation:

Obtain the values of a, b, c and d in the curve.

Since the line y=2x+1 is tangent to the curve at x=0, substitute 0 for x in y=2x+1,

y=2(0)+1y=1

Thus, the point (0,1) lies on the curve.

Substitute 0 for x and 1 for y in equation of (1),

1=04+a(0)3+b(0)2+c(0)+d1=d

Therefore, the value of d=1.

Since the line y=23x is tangent to the curve at x=1, substitute 1 for x in y=23x,

y=23(1)y=1

Thus, the point (1,1) lies on the curve.

Substitute 1 for x and -1 for y in equation of (1),

1=14+a(1)3+b(1)2+c(1)+d1=1+a+b+c+d2=a+b+c+d

Thus, the equation is a+b+c+d=2. (2)

The derivative of the curve y=x4+ax3+bx2+cx+d is dydx, which is obtained as follows.

dydx=ddx(x4+ax3+bx2+cx+d)

Apply the sum rule (3) and the constant multiple rule (1),

Apply the power rule (2) and simplify the expression,

dydx=(4x41)+a(3x31)+b(2x21)+c(1x11)+0=(4x3)+a(3x2)+b(2x)+c(1)+0=4x3+3ax2+2bx+c

Thus, the derivative of the curve is dydx=4x3+3ax2+2bx+c

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