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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Find equations of both lines through the point (2, –3) that are tangent to the parabola y = x2 + x.

(b) Show that there is no line through the point (2, 7) that is tangent to the parabola. Then draw a diagram to see why.

(a)

To determine

To find: The equation of the tangent lines to the parabola through the points.

Explanation

Given:

The parabola is y=x2+x.

The parabola through the point is (2,3).

Derivative rules:

(1) Power Rule: ddx(xn)=nxn1

(2) Sum Rule: ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x))

Formula used:

The equation of tangent line at (x1,y1) is, yy1=m(xx1) (1)

Here, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1.

Calculation:

The derivative of y is dydx, which is obtained as follows.

dydx=ddx(y) =ddx(x2+x) 

Apply the sum rule (2).

dydx=ddx(x2)+ddx(x) 

Apply the power rule (1) and simplify the expression.

dydx=2x21+1x11=2x+1

Therefore, the derivative of the parabola y=x2+x is 2x+1.

Given parabola is y=x2+x, that is every point of the parabola is (a,a2+a) of the form.

Obtain the equation of the tangent line to the parabola y=x2+x at the point (a,a2+a).

The slope of the tangent line at (a,a2+a) is,

m=dydx|x=a=2(a)+1=2a+1

Thus, the slope of tangent line at (a,a2+a) is m=2a+1 (2)

Note that, the slope of the line passing through the point (x1,x2) and (y1,y2) is m=y2y1x2x1

(b)

To determine

To show: There is no line through the point (2,7) that is tangent to the parabola.

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