Differentiation

Question

The line y=2x-16 is a tangent to the curve y=ax^3+bx at the point where x=2.Find the value of a and b.

Answer 1

y=2x-16 is the tangent line.  We know that the derivative is the slope of the tangent line.  But to find that slope, we need to find the derivative of the function in which the line is tangent to.  Basically, we are going backwards.

 

The slope of the tangent line is 2.

 

 

Starting from the tangent line, we need to find the value of y.  Plug in x=2.

 

y = 2(2) - 16

y = 4 - 16

y = -12

 

The coordinate where this line is tangent to the curve is (2, -12).

 

 

Next, we find the derivative of the curve function.  Remember that a and b are constants.

 

d/dx(ax³ + bx) = 3ax² + b

 

 

Evaluate this derivative when x=2.   The derivative is equal to 2.

 

 

3a(2)² + b = 2

 

12a + b = 2           eq1

 

 

Next, we use the coordinate we just found from the tangent line and plug in those values into the curve function.

 

-12 = 8a + 2b         eq2

 

 

We now have a system of equations.

 

12a + b = 2            eq1

 -8a - 2b = 12        eq2

 

 

Use the substitution method to solve for a and b.  Substitute eq1 into eq2.

 

-8a - 2(-12a + 2) = 12

 

-8a + 24a - 4 = 12

 

  16a = 16

 

      a = 1

 

 

Substitute this value of a into eq1 to solve for b.

 

 

(12)(1) + b = 2

 

12 + b = 2

       

        b = -10

 

a = 1

b = -10

 

 

To check, we plug in the values of a and b, find the derivative of the curve when x=2.  It should give us the derivative 2.

 

y = x³ - 10x

 

d/dx = 3x² - 10

 

d/dx = 3(2)² - 10

 

       = 12 - 10

       = 2

 

So far so good.  Next, we find the tangent line using this slope.  This is what the line looks like so far.

 

y = 2x + b

 

 

Plug in the coordinates (2, -12) to solve for b.

 

 

-12 = 2(2) + b

 

-12 = 4 + b

 

-16 = b

 

The equation of the line is

 

y = 2x - 16

 

This checks out.

 

 

Another alternative to checking is graph

 

y = 2x - 16           and           y = x³ - 10x

 

 

They should be tangent when x=2.