is perpendicular to the tangent line, so its slope will be the negative reciprocal of the tangentline’s slope. In this case, the slope of the normal line is . Finally, we plug in the point(3, 7) and the slope m = to get the equation of the normal line: y − 7 = (x − 3).- x = 9
A line that is parallel to the y-axis has an infinite (or undefined) slope. In order to find wherethe normal line has an infinite slope, we first take the derivative to find the slope of the tangentline: = 2(x − 9)(1) = 2x − 18. Next, because the normal line is perpendicular to the tangentline, the slope of the normal line is the negative reciprocal of the slope of the tangent line: m = . Now, we need to find where the slope is infinite. This is simply where the
denominator of the slope is zero: x = 9.11.
A line that is parallel to the x-axis has a zero slope. In order to find where the tangent line has azero slope, we first take the derivative: = −3 − 2x. Now we need to find where the slope iszero. The derivative −3 − 2x = 0 at x = − . Now, we need to find the y-coordinate, which weget by plugging x = − into the equation for y: 8 − 3 . Therefore, theanswer is .- a = 1, b = 0, and c = 1.
The two equations will have a common tangent line where they have the same slope, which wefind by taking the derivative of each equation. The derivative of the first equation is: = 2x +a. The derivative of the second equation is = c + 2x. Setting the two derivatives equal toeach other, we get a = c. Each equation will pass through the point (−1, 0). If we plug (−1, 0)into the first equation, we get 0 = (−1)^2 + a(−1) + b, which simplifies to: a − b = 1. If we plug