Find tangent to polynomial curve

Questions requiring differentiation of a polynomial (including those with exponential or composite factors) to find the equation of a tangent line at a specified point on the curve.

4 questions · Moderate -0.1

1.07m Tangents and normals: gradient and equations

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OCR C1 Q10

14 marks Moderate -0.3

10. The curve $C$ has the equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = ( x + 2 ) ^ { 3 }$$

Sketch the curve $C$, showing the coordinates of any points of intersection with the coordinate axes.
Find $\mathrm { f } ^ { \prime } ( x )$. The straight line $l$ is the tangent to $C$ at the point $P ( - 1,1 )$.
Find an equation for $l$. The straight line $m$ is parallel to $l$ and is also a tangent to $C$.
Show that $m$ has the equation $y = 3 x + 8$.

OCR C1 Q10

13 marks Moderate -0.3

10. The curve with equation $y = ( 2 - x ) ( 3 - x ) ^ { 2 }$ crosses the $x$-axis at the point $A$ and touches the $x$-axis at the point $B$.

Sketch the curve, showing the coordinates of $A$ and $B$.
Show that the tangent to the curve at $A$ has the equation $$x + y = 2$$ Given that the curve is stationary at the points $B$ and $C$,
find the exact coordinates of $C$.

OCR C1 2011 June Q10

16 marks Standard +0.3

10 A curve has equation $y = ( 2 x - 1 ) ( x + 3 ) ( x - 1 )$.

Sketch the curve, indicating the coordinates of all points of intersection with the axes.
Show that the gradient of the curve at the point $P ( 1,0 )$ is 4 .
The line $l$ is parallel to the tangent to the curve at the point $P$. The curve meets $l$ at the point where $x = - 2$. Find the equation of $l$, giving your answer in the form $y = m x + c$.
Determine whether $l$ is a tangent to the curve at the point where $x = - 2$.

Edexcel C1 Q10

14 marks Moderate -0.3

10. The curve $C$ has the equation $y = \mathrm { f } ( x )$ where $$f ( x ) = ( x + 2 ) ^ { 3 }$$

Sketch the curve $C$, showing the coordinates of any points of intersection with the coordinate axes.
Find f ${ } ^ { \prime } ( x )$. The straight line $l$ is the tangent to $C$ at the point $P ( - 1,1 )$.
Find an equation for $l$. The straight line $m$ is parallel to $l$ and is also a tangent to $C$.
Show that $m$ has the equation $y = 3 x + 8$.