Question

calculate the values of a and b if f(x) = ax^2 + bx + 5 has a tangent at x = -1 which is defined by equation y = -7x + 3. Please help...

Answer 1

Question:

Calculate the values of a and b if f(x) = ax²+bx+5 has a tangent at x = -1 which is defined by equation y = -7x + 3.

Answer:

a = 2 , b = -3

Note:

• The first derivative of the equation of any curve at a point will give the slope of the tangent at that point.

• The slope,y-intercept form of a straight line is given as ; y = mx + c , where m is the slope and c is the y-intercept of the straight line.

Solution:

The given equation of the curve is ;

f(x) = ax² + bx + 5 , ie ; y = ax² + bx + 5 -----(1)

Now,

Differentiating both sides of eq-(1) wrt x , we get ;

=> dy/dx = d(ax²)/dx + d(bx)/dx + d(5)/dx

=> dy/dx = 2ax + b

=> (dy/dx) at x = -1 = 2a•(-1) + b

=> (dy/dx) at x = -1 = -2a + b -------(2)

Now,

The given equation of tangent is ;

y = -7x + 3 --------(3)

Comparing eq-(3) with the the equation y = mx + c we get ;

Slope , m = -7

y-intercept , c = 3

We know that,

The first derivative of the equation of any curve at a point gives the slope of the tangent at that point.

Thus,

=> (dy/dx) at x = -1 of the curve is equal to the slpoe (m) of the tangent line.

=> -2a + b = m

=> -2a + b = - 7 -------(4)

Now,

Let's find the y-coordinate ( of the point of contact of the curve and the tangent line ) at x = -1 using eq-(1) ;

=> y = ax² + bx + 5

=> y = a(-1)² + b(-1) + 5

=> y = a - b + 5 ------(5)

Now,

Let's find the y-coordinate ( of the point of contact of the curve and the tangent line ) at x = -1 using eq-(3) ;

=> y = -7x + 3

=> y = -7(-1) + 3

=> y = 7 + 3

=> y = 10 ------(6)

Also,

We know that, the tangent touches the curve at a single point , thus the y-coordinate of the curve and the tangent line would be equal.

Hence,

From eq-(5) and eq-(6) , we get ;

=> a - b + 5 = 10

=> a - b = 10 - 5

=> a - b = 5 ---------(7)

Now,

Adding eq-(4) and eq-(7) , we get ;

=> -2a + b + a - b = -7 + 5

=> -a = -2

=> a = 2

Now,

Putting a = 2 in eq-(7) , we get ;

=> a - b = 5

=> 2 - b = 5

=> - b = 5 - 2

=> - b = 3

=> b = - 3 .

Hence,

The required values of a and b are 2 and -3 respectively .