IB Maths AI HL AI 5.1 Differentiation questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

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Question 1

Paper 2Medium14 marks

A small mass moves vertically on a spring. For $t\geq 0$ its displacement $y$ metres from its equilibrium position is modelled by

y(t)=0.32\sin(1.4t+0.30),

where $t$ is measured in seconds and angles are in radians. This is a model of simple harmonic motion.

(a)

State the amplitude and period of the motion. Calculate the displacement of the mass at $t=0$ giving your answer to three significant figures.

[3]

(b)

Show that

\frac{d^2y}{dt^2}=-0.6272\sin(1.4t+0.30),

and hence show that

\frac{d^2y}{dt^2}=-1.96y(t).

[3]

(c)

Determine the time of the first stationary point after $t=0$ Use the second derivative test to classify this point.

[4]

(d)

Using your result from part (b), determine the time of the first point of inflexion after $t=0$ State the concavity immediately before and after this point, and interpret the point in terms of the mass's motion.

[4]

Question 2

Paper 2Medium13 marks

A landscape designer models the vertical profile of a section of a footpath by

h(x)=0.009x^3-0.25x^2+1.7x+2.5,\qquad 0\leq x\leq20,

where $x$ is the horizontal distance, in metres, from the start of the section and $h(x)$ is the height, in metres, above a reference level.

Consider the point on the profile where $x=8$

(a)

Calculate the height of the profile at this point.

[2]

(b)

Show that

h'(x)=0.027x^2-0.50x+1.7.

[2]

(c)

Determine the equation of the tangent to the profile at $x=8$ giving your answer in the form $h-h(8)=m(x-8)$

[3]

(d)

Determine the equation of the normal to the profile at $x=8$ in the same form. Give its gradient to 3 significant figures.

[3]

(e)

Use your GDC to find the coordinates of the other point, different from the point where $x=8$ at which the tangent in part (c) meets the profile. Give your coordinates to 3 significant figures.

[3]

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