IB · Mathematics: Applications & Interpretation SL & HL · Calculus

Differentiation & tangents: IB Mathematics: Applications & Interpretation questions with answers

Free Differentiation & tangents questions for IB Mathematics: Applications & Interpretation (SL & HL), part of Calculus. Each question has the answer and the marking points an examiner would look for.

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Differentiation & tangents practice questions

Question 1Structured · 7 marksA* challenge

Consider the curve y = x2 + 54x, x ≠ 0.

(a)

Find dydx.

[2]
(b)

Find the equation of the tangent to the curve at x = 1, in the form y = mx + c.

[3]
(c)

The tangent meets the x-axis at A and the y-axis at B. Find the area of triangle OAB.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2x − 54x2
2x A1
−54x2 A1
2
(b)y = −52.0x + 107
gradient −52.0 (A1)
point (1, 55.0) (A1)
y = −52.0x + 107 A1
3
(c)110
A(2.06, 0), B(0, 107) (M1)
110 A1
2
Question 2Structured · 5 marks

The profit of a company is P(x) = −2x3 + 48x + 10, where x is the number of thousands of items sold.

(a)

Find P′(x).

[2]
(b)

Find the number of items that maximizes the profit and the maximum profit.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)−6x2 + 48
power rule A1
−6x2 + 48 A1
2
(b)2828 items; 101
P′(x) = 0 (M1)
x = 2.83 A1
101 A1
3
Question 3Structured · 6 marks

Let f(x) = x3 + 4x2, x ≠ 0.

(a)

Find f′(x).

[2]
(b)

Find the equation of the tangent to the graph of f at the point where x = 2.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)3x2 − 8x−3
use of x−2 (M1)
3x2 − 8x−3 A1
2
(b)y = 11x − 13
f(2) = 9.00 (A1)
f′(2) = 11.0 (A1)
y − 9.00 = 11.0(x − 2) (M1)
y = 11.0x − 13.0 A1
4
Question 4Structured · 7 marksA* challenge

Consider the curve y = x2 + 27x, x ≠ 0.

(a)

Find dydx.

[2]
(b)

Find the equation of the tangent to the curve at x = 1, in the form y = mx + c.

[3]
(c)

The tangent meets the x-axis at A and the y-axis at B. Find the area of triangle OAB.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2x − 27x2
2x A1
−27x2 A1
2
(b)y = −25.0x + 53.0
gradient −25.0 (A1)
point (1, 28.0) (A1)
y = −25.0x + 53.0 A1
3
(c)56.2
A(2.12, 0), B(0, 53.0) (M1)
56.2 A1
2
Question 5Structured · 5 marks

The profit of a company is P(x) = −3x3 + 27x + 10, where x is the number of thousands of items sold.

(a)

Find P′(x).

[2]
(b)

Find the number of items that maximizes the profit and the maximum profit.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)−9x2 + 27
power rule A1
−9x2 + 27 A1
2
(b)1732 items; 41.2
P′(x) = 0 (M1)
x = 1.73 A1
41.2 A1
3
Question 6Structured · 6 marks

Let f(x) = x3 + 6x2, x ≠ 0.

(a)

Find f′(x).

[2]
(b)

Find the equation of the tangent to the graph of f at the point where x = 1.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)3x2 − 12x−3
use of x−2 (M1)
3x2 − 12x−3 A1
2
(b)y = −9x + 16
f(1) = 7.00 (A1)
f′(1) = −9.00 (A1)
y − 7.00 = −9.00(x − 1) (M1)
y = −9.00x + 16.0 A1
4

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in