Tangent and Normal

The equations of the normal to the curve  $x=1-\cos \theta :y=\theta -\sin \theta$ at $\theta =\frac{\pi }{4}$  would be:

The equation of a tangent to the curve $y=\sqrt{3x-2}$ which is parallel to the line $4x-2y+5=0$ would be

Let the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}, \left(a>0\right)$ at any point on it cuts the coordinate axes at $P$ and $Q.$ Then $OP + OQ,$ where $O$ is origin, is equal to

Find the area of the triangle formed by the $x$-axis and the tangent and the normal to the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\left(\frac{a}{\sqrt{2}}, \frac{b}{\sqrt{2}}\right)$

If the normal to the curve $y=f\left(x\right)$ at the point $\left(3,4\right)$ makes an angle $\frac{3\pi }{4}$ with the positive $x$-axis, then $f^{\prime }(3)$ is equal to

The tangent at the point$(2,-2)$ to the curve, $x^2{y}^2-2x=4\left(1-y\right)$ does not pass through the point:

The gradient of the tangent to the function with equation $y=a{x}^2+bx+3$ at the point $(2,-1)\text{ is }8$. Find the value of $a$ and $b$.

A curve is given by the equation $y=\frac{2x^3}{3}-\frac{7x^2}{2}+2x+5$. Determine the coordinates on the curve where the gradient is $-3$. You must show all your working, and give your answers as exact fractions.

Jacek is practising on his skateboard. His journey along the track can be modelled by a curve with equation $f\left(t\right)=10.667-1.667t+0.0417t^2$, where $t$ is the time in seconds and $f\left(t\right)$ is the distance in metres. Find $f^{\text{'}}\left(12\right)\text{ and }{f}^{\text{'}}\left(32\right)$ and comment on these values.

The gradient of the normal to the curve $f\left(x\right)=3x^2+4x-3$ at point $B\text{ is }\frac{1}{2}$ Find the coordinates of point $B$.

The gradient of the tangent to the curve $f\left(x\right)=x^2-5x-4$ at point $A$ is $1$ .Find the coordinates of point $A$.

Find the derivative of the following function with respect to $x$ and find the value of the gradient of the tangent to the curve at $x=1$,

$f\left(x\right)=\frac{6}{x^3}+2x^2+8$.

Find the derivative of the following function with respect to $x$ and find the value of the gradient of the tangent to the curve at $x=1$,

$f\left(x\right)=\frac{2}{x}+3x-9$.

Find the derivative of the following function with respect to $x$ and find the value of the gradient of the tangent to the curve at $x=1$,

$y=6x^3-3x^2+x-7$.

Find the derivative of the following function with respect to $x$ and find the value of the gradient of the tangent to the curve at $x=1$,

$g\left(x\right)=2x^2-4x-1$.

Find the derivative of the following function with respect to $x$ and find the value of the gradient of the tangent to the curve at $x=1$,

$y=3x+5$.

Find the derivative of the following function with respect to $x$ and find the value of the gradient of the tangent to the curve at $x=1$,

$f\left(x\right)=6$.

Find the gradient of the following curve at the point where $x=3$,

$g\left(x\right)=\left(x^2-3\right)(x-1{)}^5$.

Find the gradient of the following curve at the point where $x=3$,

$y=x{e}^{2x}$[ Write approximate value correcting to only numerical value, use $e^6=403.428793$]

Find the gradient of the following curve at the point where $x=3$,

$y=\frac{x+5}{x^3+1}$.