Question Type 2: Calculating the normal to the curve at the point where tangent is known Exercises

Calculate the equation of the normal to the curve $y = x^2 + 2x + 1$ at the point where the tangent is parallel to the line $y = 4x + 3$.

For the curve $y = 3x^3 - x$, find the equations of the normals at the points where the tangent is horizontal.

Find the equation of the normal to $y = 4x^4 - 8x^2 + 5$ at $x = 1$.

The normal to the curve $y = x^2 + kx + 3$ at $x = 3$ has gradient $-\frac{1}{4}$. Find the value of $k$ and the equation of this normal.

Find the equation of the normal to the curve $y=x^5-5x+4$ at $x=2$.

Find the equation of the normal to the curve $y = x^4 - 2x + 1$ at $x = 1$.

For the curve $y = \frac{1}{x}$, find the equation(s) of the normal(s) at the point(s) where the tangent is parallel to the line $y = -x + 4$.

The normal to $y = x^3 - ax^2 + bx - 1$ at $x = 1$ is $y = x + 2$. Find the values of $a$ and $b$.

The tangent to the curve $y = 2x^3 + x^2 - 4x + 1$ is $y = 4x - 4$. Find the equation of the normal at the point of contact.

Find the equation of the normal to the curve $y = x^2 - 4x + 7$ at $x = 2$.

Find the equations of the normals to the curve $y = x^3 - 3x + 2$ at the points where the tangents are parallel to the line $y = 9x - 7$.

Find the equation of the normal to the curve $y = 3x^3 - x^2 + 2$ at the point where $x = -1$.