Find constants from gradient facts

Find unknown constants from gradient and point facts

In 4 of 20 recent exams
Paper 1 or 2
2 to 6 marks

How it is set in the exam

Paper 1 (no calculator) (Paper 2 (calculator) with a GDC system), 2–6 marks. Typical.

Spot it in the exam

Exam phrases to spot (4)

“the tangent at $x = 2$ is $y = 7x - 10$ ; find $a$ and $b$ ”
“the gradient of the curve at $(2, 5)$ is $3$ ”
“ $f$ has a local minimum at $(2, -1)$ ; find $a$ and $b$ ”
“find the value of $p$ and the value of $q$ ”

In disguise (3)

"the curves touch at $x = 1$ ": equal values and equal gradients there
"a turning point on the y-axis": $f'(0) = 0$
"the normal at $x = 2$ is parallel to …": convert to the tangent's gradient first

Looks like it, but isn't (3)

$f'(x)$ is given, with a point on the graph: integrate (Find f from f′ and a point
the constants are known and the point is wanted: Find the point of contact
a line with an unknown is a tangent to a parabola, and no point is given: Two graphs

Worked example

One tangent line, two facts

Paper 1 (no calculator)

Question

The line $y = 7x - 10$ is a tangent to the curve $y = ax^2 + \dfrac{b}{x}$ at the point where $x = 2$ . Find the value of $a$ and the value of $b$ . [6]

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Steps

Count
the unknowns and the facts. Split every compound fact into its equations (table below): "the tangent at $x = 2$ is $y = 7x - 10$ " is two facts.
Write one equation per fact.
Differentiate first if a fact is about a gradient. When a tangent line is given, take the point's $y$ from the line.
Solve
, starting with the equation that has the fewest unknowns. On , write every equation down, then use the GDC (solve simultaneous linear equations).

Check

by substituting back into every fact.

Fact	Equation(s)
$(x_1, y_1)$ is on the curve	$f(x_1) = y_1$
the gradient at $x = x_1$ is $m$	$f'(x_1) = m$
the tangent at $x = x_1$ is $y = mx + c$	$f'(x_1) = m$ and $f(x_1) = mx_1 + c$ (two facts)
the normal at $x = x_1$ has gradient $n$	$f'(x_1) = -\frac{1}{n}$
two curves touch (share a tangent) at $x = x_1$	$f(x_1) = g(x_1)$ and $f'(x_1) = g'(x_1)$
a stationary point (turning point, maximum, minimum) at $(x_1, y_1)$	$f'(x_1) = 0$ and $f(x_1) = y_1$
a point of inflexion at $x = x_1$	$f''(x_1) = 0$

Common mistakes

Using a tangent line as one fact.
"The tangent at $x = 2$ is $y = 7x - 10$ " gives two equations: the point and the gradient.
Taking the $y$ -value from the curve
before the constants are known. When the tangent line is given, the point's $y$ comes from the line.
Using only $f'(x_1) = 0$ for a stationary point $(x_1, y_1)$ .
$f(x_1) = y_1$ is the second equation.

More mistakes (3)

Differentiating a constant term $b$ as if it were $bx$ .
It differentiates to $0$ .
Not converting a normal's gradient.
A normal gradient $n$ means $f'(x_1) = -\frac{1}{n}$ .
Rounding $a$ and $b$
before using them in a later part ().

Easy to confuse with

Next typeThe tangent or normal meets something

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