IB Mathematics Analysis and Approaches Topic AHL 5.12—first Principles, Higher Derivatives Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 5.12—first Principles, Higher Derivatives Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 5.12—first Principles, Higher Derivatives with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.12—first Principles, Higher Derivatives and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 2

In physics, the displacement of a particle in a damped oscillatory system is modeled by $h (t) = e^{- t} sin (2 t)$ , where $t$ is time in seconds (radians are assumed for trigonometric functions).

Find $h^{'} (t)$ using first principles.

[8]

Show that $h (t)$ satisfies the differential equation $h^{''} (t) + 2 h^{'} (t) + 5 h (t) = 0$ .

[6]

Find the Maclaurin series for $h (t)$ up to the $t^{5}$ term.

[6]

Determine the coordinates of the first positive maximum of $h (t)$ .

[4]

Question 2

HLPaper 1

Let $f (x) = tan x$ , for $x \in (- \frac{π}{2}, \frac{π}{2})$ .

Find $f^{'} (x)$ using first principles.

[6]

Compute the first three derivatives of $f (x)$ .

[5]

Find the Taylor polynomial of degree $3$ for $f (x)$ centered at $x = 0$ .

[5]

Question 3

HLPaper 2

Let $p (x) = x^{2} cos x$ .

Find $p^{'} (x)$ using first principles.

[6]

Find $p^{''} (x)$ . , and sketch $y = p (x)$ on $[0, π]$ , labeling these points.

[2]

Solve $p^{''} (x) = 0$ for $x \in [0, π]$ , giving approximate solutions.

[3]

Find the equation of the normal line to $p (x)$ at $x = \frac{π}{2}$ .

[4]

Question 4

HLPaper 1

Let $h (x) = x^{4} - 2 x^{2}$ .

Find $h^{'} (x)$ using first principles.

[5]

Find the $x$ -coordinates of the points of inflection (where concavity changes).

[4]

Find the equation of the normal line to the curve at $x = 1$ .

[4]

Question 5

HLPaper 1

Let $f (x) = 2 x^{2} + 3 x$ .

Find the derivative of $f (x)$ using first principles.

[5]

Find the second and third derivatives of $f (x)$ .

[3]

Find the equation of the tangent to $y = f (x)$ at $x = 1$ . Sketch the curve and the tangent line on the same axes, clearly labeling the point of tangency.

[3]

IB Mathematics Analysis and Approaches AHL 5.12—first Principles, Higher Derivatives Questionbank

Unlock all 35 questions with a free account

IB Mathematics Analysis and Approaches AHL 5.12—first Principles, Higher Derivatives Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 5.12—first Principles, Higher Derivatives Questionbank

Unlock all 35 questions with a free account