Q1
Answer
Find the gradient of $y = 2 - 3x^2$ at the point where $x=4$.
$-24$
Q2
Answer
Find the gradient of $y = 4x^3 - x$ at the point where $x=2$.
$47$
Q3
Answer
Given $y = 2x^2 - 5x + 1$, find the $x$ coordinate when the curve has a gradient of $7$.
$x = 3$
Q4
Answer
Find the coordinates of the points on the curve $y = x^3 - 8x$ which have a gradient of $4$.
$(2, -8)$ and $(-2, 8)$
Q5
Answer
Find the equation of the tangent to the curve $y = 2x^2 - 6x + 8$ at the point where $x = 1$.
$y = -2x + 6$
Q6
Answer
Find the equation of the normal to the curve $y = x^3 - 8x + 4$ at the point where $x = 2$.
$x + 4y + 14 = 0$
Q7
Answer
Find the equation of the tangent to the curve $y = 3x^2 - 5x + 2$ at the point where $x = 2$.
$y = 7x - 10$
Q8
Answer
Find the equation of the normal to the curve $y = x^3 + 5x^2 - 12$ at the point where $x = -3$.
$y = \dfrac{1}{3}x + 7$
Q9
Answer
A curve has equation $y = x^2 - 3x + 4$.
  1. Find an equation of the normal to the curve at the point $A(2,2)$.
  2. The normal to the curve intersects the curve again. Find the coordinates of the other point of intersection.
  1. $y = 4 - x$
  2. $(0,4)$
Q10
Answer
The line $y = 2x + k$ is a normal to the curve with equation $y = \dfrac{16}{x^2}$. Find $k$.
$k = -7$
Q11
Answer
A curve has equation $y = 2 + 3x + kx^2 - x^3$. The gradient is $-6$ at the point $P$ where $x=-1$.
  1. Find $k$.
  2. The tangent to the curve at the point $Q$ is parallel to the tangent at $P$. Find the length $PQ$ in the form $a\sqrt{5}$.
  1. $k = 3$
  2. $4\sqrt{5}$
Q12
Answer
A curve has equation $y = 2x^2 - 7x + 1$. The point $A$ has $x$ coordinate $2$.
  1. Find an equation of the tangent to the curve at $A$.
  2. The normal to the curve at point $B$ is parallel to the tangent at $A$. Find the coordinates of $B$.
  1. $y = x - 7$
  2. $(1.5, -5)$
Q13
Answer
A curve has equation $y = \sqrt{x}(k - x)$. The gradient of the curve is $\sqrt{2}$ at the point $P$ where $x=2$.
  1. Find $k$.
  2. Show that the normal to the curve at $P$ has the equation $x + \sqrt{2}y = c$.
  1. $k = 10$
  2. $x + \sqrt{2}y = 18$
Q14
Answer
A curve has equation $y = x^3 - 3x^2 - 8x + 4$. The line $l$ is tangent to the curve at the point $P(-1,8)$.
  1. Find an equation for $l$.
  2. The line $m$ is parallel to $l$ and is tangent to the curve at a different point, $Q$. Find an equation for $m$.
  1. $y = x + 9$
  2. $y = x - 23$
Q15
Answer
Find the range of values for $x$ such that $y = 3x^2 + 8x + 2$ is increasing.
$x > -\dfrac{4}{3}$
Q16
Answer
Find the range of values for $x$ such that $y = 4x - 3x^2$ is increasing.
$x < \dfrac{2}{3}$
Q17
Answer
Find the range of values for $x$ such that $y = 5 - 8x - 2x^2$ is increasing.
$x < -2$
Q18
Answer
Find the range of values for $x$ such that $y = x^4 - 8x^3$ is increasing.
$x > 6$
Q19
Answer
Find the range of values for $x$ such that $y = 1 - 27x + x^3$ is decreasing.
$-3 < x < 3$
Q20
Answer
Find the range of values for $x$ such that $y = x + \dfrac{25}{x}$ is decreasing.
$-5 < x < 5$
Q21
Answer
Find the range of values for $x$ such that $y = x^2(x + 3)$ is decreasing.
$-2 < x < 0$
Q22
Answer
Giventhat $y = 2x^3 + 3x^2 + kx + 5$ is decreasing when $- 2 < x < 1$, find $k$.
$-12$
Q23
Answer
Given $f(x) = 4 - x(2x^2 + 3)$, show that $f$ is decreasing for all values of $x$.
$x^2$ is always positive
Q24
Answer
The function $f(x) = x^2 + px$ is increasing in the range $-1 \leq x \leq 1$. Find one possible value for $p$.
$p > 2$
Q25
Answer
Sketch the gradient function of this curve:

Q26
Answer
Sketch the gradient function of this curve:

Q27
Answer
Sketch the gradient function of this curve:

Q28
Answer
Sketch the gradient function of this curve: