Find tangent given derivative expression - Tangents, normals and gradients

Edexcel C1 2012 June Q7

8 marks Moderate -0.3

7. The point $P ( 4 , - 1 )$ lies on the curve $C$ with equation $y = \mathrm { f } ( x ) , x > 0$, and $$f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - \frac { 6 } { \sqrt { } x } + 3$$

Find the equation of the tangent to $C$ at the point $P$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are integers.
Find $\mathrm { f } ( x )$.

Edexcel C1 2017 June Q7

9 marks Standard +0.3

7. The curve $C$ has equation $y = \mathrm { f } ( x ) , x > 0$, where $$\mathrm { f } ^ { \prime } ( x ) = 30 + \frac { 6 - 5 x ^ { 2 } } { \sqrt { x } }$$ Given that the point $P ( 4 , - 8 )$ lies on $C$,

find the equation of the tangent to $C$ at $P$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants.
Find $\mathrm { f } ( x )$, giving each term in its simplest form.

OCR C2 Q9

13 marks Standard +0.3

9. The curve $C$ has the equation $y = \mathrm { f } ( x )$ where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line $l$ has the equation $y = 2 x - 1$ and is a tangent to $C$ at the point $P$.

State the gradient of $C$ at $P$.
Find the $x$-coordinate of $P$.
Find an equation for $C$.
Show that $C$ crosses the $x$-axis at the point $( 1,0 )$ and at no other point.

AQA C1 2013 January Q6

8 marks Moderate -0.5

6 The gradient, $\frac { \mathrm { d } y } { \mathrm {~d} x }$, of a curve at the point $( x , y )$ is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 10 x ^ { 4 } - 6 x ^ { 2 } + 5$$ The curve passes through the point $P ( 1,4 )$.

Find the equation of the tangent to the curve at the point $P$, giving your answer in the form $y = m x + c$.
Find the equation of the curve.

Edexcel C1 Q9

13 marks Standard +0.3

9. The curve $C$ has the equation $y = \mathrm { f } ( x )$ where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line $l$ has the equation $y = 2 x - 1$ and is a tangent to $C$ at the point $P$.

State the gradient of $C$ at $P$.
Find the $x$-coordinate of $P$.
Find an equation for $C$.
Show that $C$ crosses the $x$-axis at the point $( 1,0 )$ and at no other point.

CAIE P1 2019 March Q2

5 marks Moderate -0.3

A curve with equation $y = f(x)$ passes through the points $(0, 2)$ and $(3, -1)$. It is given that $f'(x) = kx^2 - 2x$, where $k$ is a constant. Find the value of $k$. [5]

CAIE P1 2016 November Q10

12 marks Standard +0.3

A curve is such that $\frac{dy}{dx} = \frac{2}{a}x^{-\frac{1}{2}} + ax^{-\frac{3}{2}}$, where $a$ is a positive constant. The point $A(a^2, 3)$ lies on the curve. Find, in terms of $a$,

the equation of the tangent to the curve at $A$, simplifying your answer, [3]
the equation of the curve. [4]

It is now given that $B(16, 8)$ also lies on the curve.

Find the value of $a$ and, using this value, find the distance $AB$. [5]