CAIE A-Level Mathematics AS 1.7 Differentiation Questions

Practise differentiating powers and composite expressions and applying derivatives to gradients, tangents, normals, stationary points, monotonicity and connected rates.

Exam points

apply power and chain rules and simplify first or second derivatives into usable form
set dy/dx to a gradient or zero to find tangents, normals and stationary points
connect rates with dy/dt = (dy/dx)(dx/dt) and substitute values with consistent units

Covered syllabus points

1.7.1—Gradient as limit 1.7.2—Differentiation rules 1.7.3—Differentiation applications 1.7.4—Stationary points

[Maximum number: 3]

The equation of a curve is y=f(x), where $\mathrm{f}(x)=(2 x-1) \sqrt{3 x-2}-2$ . The following points lie on the curve. Non-exact values have been given correct to 5 decimal places.

A(2,4), B(2.0001, k), C(2.001,4.00625), D(2.01,4.06261), E(2.1,4.63566), F(3,11.22876)

(a)

Find the value of k. Give your answer correct to 5 decimal places.

The table shows the gradients of the chords A B, A C, A D and A F.

Table for Question (a) — CAIE A-Level Mathematics AS

[ 1 ]

(b)

Find the gradient of the chord A E. Give your answer correct to 4 decimal places.

[ 1 ]

(c)

Deduce the value of $f^{\prime}(2)$ using the values in the table.

[ 1 ]

Suggests that $[\mathrm{f}(2)]=6.25$
CAO

[Maximum number: 8]

The equation of a curve is

y=k \sqrt{4 x+1}-x+5,

where k is a positive constant.

(a)

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

[ 2 ]

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\left\{k \frac{1}{2}(4 x+1)^{-\frac{1}{2}}\right\}\{\times 4\}\{-1\}$

B2,1,0

OE e.g. $2 k(4 x+1)^{-\frac{1}{2}}-1$

B2 Three correct unsimplified \{ \} and no others.

B1 Two correct $\}$ or three correct $\}$ and an additional term e.g. +5.

B0 More than one error.

(b)

Find the x-coordinate of the stationary point in terms of k.

[ 2 ]

$2 k(4 x+1)^{-\frac{1}{2}}-1=0$ leading to $(4 x+1)^{\frac{1}{2}}=2 k$ or $\frac{2 k}{(4 x+1)^{\frac{1}{2}}}=1$
OE Equating their $\frac{d y}{d x}$ of the from $a k(4 x+1)^{-\frac{1}{2}}-1$ where
a=2 or 0.5 , to 0 and dealing with the negative power
correctly including k not multiplied by $(4 x+1)^{\frac{1}{2}}$ .

$x=\frac{4 k^{2}-1}{4}$
CAO

OE simplified expression ISW.

(c)

Given that k=10.5, find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of $\tan ^{-1}(2)$ with the positive x-axis.

[ 4 ]

$2 \times 10.5(4 x+1)^{-\frac{1}{2}}-1=2$
Putting k=10.5 into their $\frac{d y}{d x}$ and equating to 2.

$7=(4 x+1)^{\frac{1}{2}}$ leading to 4 x+1=49 leading to x=12
If M1 earned SCB1 available for $x=\frac{33}{64}$ from $a=\frac{1}{2}$ .

$y=[10.5 \sqrt{4 x+1}-x+5=] 66.5$ [leading to (12, 66.5)]
$y-66.5=-\frac{1}{2}(x-12)$
OE

[Maximum number: 7]

The equation of a curve is $y=2 x^{2}-\frac{1}{2 x}+3$ .

(a)

Find the coordinates of the stationary point.

[ 3 ]

Differentiate to obtain $4 x+\frac{1}{2} x^{-2}$
OE
Condone ' +c '.

Equate first derivative to zero and solve $4 x+\frac{K}{x^{2}}=0$ as far as $x^{3}=k, K$ and k nonzero
Equate first derivative to zero and solve $4 x+\frac{K}{x^{2}}=0$ as far as $x^{3}=k, K$ and k non-
zero
Not given if ' +c ' used.

$x=-\frac{1}{2}$ and $y=\frac{9}{2}$
OE
B1 SC if no visible solution of the cubic.

(b)

Determine the nature of the stationary point.

[ 2 ]

Differentiate their first derivative, substitute their x value. Substitution may be
implied by a correct inequality or correct value,
Must differentiate one term correctly.

Expect $4-x^{-3}=12$ at $x=\frac{-1}{2}$

Alternative: substitute values of x into $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

One value $x<-\frac{1}{2}$ and one value $-\frac{1}{2}<x<0$ .

conclude minimum
Following correct work only

(c)

For positive values of x, determine whether the curve shows a function that is increasing, decreasing or neither. Give a reason for your answer.

[ 2 ]

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question (c)

Question 3

Question (a)

Question (b)

Question (c)