IB Maths AA HL Sl 5 4 Tangents and Normals Questions

Practise finding gradients, equations and interpretations for tangents and normals at a point, using analytic methods or technology where appropriate.

Exam points

Find the gradient of a tangent or normal from a derivative and a specified point or input, including gradients obtained by chain, product, quotient or implicit differentiation.
Use the point-slope form to determine the equation of a tangent or normal and solve for its intercept, intersection or another point on the line.
Use equal, parallel or perpendicular gradients to determine unknown points, parameters or tangent/normal relationships.
Prove tangent relationships, including horizontal tangents and perpendicular intersections, from derivative equations or a family of curves.

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Study SL 5.4—Tangents and normals conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 6]

Consider the function f defined by $f(x)=90 \mathrm{e}^{-0.5 x}$ for $x \in \mathbb{R}^{+}$ .
The graph of f and the line y=x intersect at point P .

(a) attempt to find the point of intersection of the graph of f and the line y=x (M1)
$x=5.56619 \ldots$
=5.57

(a)

Find the exact coordinates of Q.

[ 4 ]

$\quad f^{\prime}(x)=-45 \mathrm{e}^{-0.5 x}$
attempt to set the gradient of f equal to -1
$-45 \mathrm{e}^{-0.5 x}=-1$
Q has coordinates $(2 \ln 45,2)$ (accept $\left(-2 \ln \frac{1}{45}, 2\right)$

Note: Award A1 for each value, even if the answer is not given as a coordinate pair.

Do not accept $\frac{\ln \frac{1}{45}}{-0.5}$ or $\frac{\ln 45}{0.5}$ as a final value for x. Do not accept 2.0 or 2.00 as a final value for y.

(b)

Show that the equation of L is $y=-x+2 \ln 45+2$ .

The shaded region A is enclosed by the graph of f and the lines y=x and L.

Figure for Question (b) — IB Maths AA HL

[ 2 ]

attempt to substitute coordinates of Q ( in any order)
into an appropriate equation
$y-2=-(x-2 \ln 45)$ OR $2=-2 \ln 45+c$
equation of L is $y=-x+2 \ln 45+2$

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IB Maths AA HL Sl 5 4 Tangents and Normals Questions question 1

Question (a)

Question (b)