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SL 5.4—Tangents and normal - IB Questionbank | RevisionDojo

Practice SL 5.4—Tangents and normal with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider the function $g(x) = e^x \cos(x)$ ,.

Find the derivative of the function $g(x) = e^x \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Find the equation of the normal line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Question 2

SLPaper 1

The function j is defined for all x ∈ ℝ. The line with equation y = 6x - 1 is the tangent to the graph of j at x = 4.

Write down the value of j'(4).

[1]

Find j(4).

[1]

The function k is defined for all x ∈ ℝ where k(x) = x² - 3x and m(x) = j(k(x)). Find m(4).

[2]

Hence find the equation of the tangent to the graph of m at x = 4.

[3]

Question 3

SLPaper 1

Let $g(x) = \sin(x) + \cos(x)$ .

Find the derivative of the function $g(x)$ .

[1]

Find the equation of the tangent line to the graph of $g(x)$ at $x = \frac{\pi}{4}$ .

[3]

Hence, or otherwise, show that at $x=\frac{\pi}{4}$ the function attains its maximum.

[4]

Question 4

HLPaper 1

Consider the function $f(x) = x^2 \sin(x) + \cos(x)$ .

Find the derivative of the function $f(x) = x^2 \sin(x) + \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $f(x) = x^2 \sin(x) + \cos(x)$ at the point where $x = \frac{\pi}{2}$ .

[2]

Find the equation of the normal line to the curve $f(x) = x^2 \sin(x) + \cos(x)$ at the point where $x = \frac{\pi}{2}$ .

[2]

Question 5

SLPaper 1

Consider the path with equation $y=(2x-1)e^{kx}$ , where $x\in\mathbb{R}$ and $k\in\mathbb{Q}$ .

The tangent to the path at the point where $x=1$ is parallel to the line $y=5e^{k}x$ .

Find the value of $k$ .

[5]

Question 6

SLPaper 2

The curve $C$ is defined by the function $g(x)=2 x^{4}-8 x^{2}+5$ . The tangent to $C$ at the point $P$ where $x=1$ is parallel to the line $y=-8 x+3$ .

Find the gradient of the tangent to $C$ at $x=1$ .

[3]

Determine the equation of the tangent at $P$ .

[3]

Sketch the graph of $y=g(x)$ and the tangent at $P$ .

[3]

Question 7

SLPaper 2

The function $f(x)=20 e^{-0.2 x}$ for $x \in \mathbb{R}^{+}$ intersects the line $y=2 x+3$ at point $P$ . The tangent to $f$ at point $Q$ has a gradient of -2 . The line $L$ is the normal to $f$ at $Q$ , intersecting the line $y=2 x+3$ at point $R$ . The function $h(x)=f\left(x^{2}-2 x+1\right)$ is defined for all $x \in \mathbb{R}$ .

Find the exact coordinates of $P$ .

[3]

Determine the exact coordinates of $Q$ .

[3]

Show that the equation of $L$ is $y=\frac{1}{2} x+\ln 10+\frac{1}{2}$ .

[3]

Find the coordinates of $R$ .

[3]

Find the area of the region enclosed by $f, y=2 x+3$ , and $L$ .

[5]

Find $h^{\prime}(x)$ and determine the gradient of the tangent to $h$ at $x=1$ .

[4]

Question 8

SLPaper 1

The function $f(x)=\frac{1}{4} x^{4}-2 x^{2}+3$ has horizontal tangents at points $A$ and $B$ , where $x=a$ and $x=b, a<b$ . The normal at $x=a$ and the tangent at $x=b$ intersect at point $P(p, q)$ .

Find $f^{\prime}(x)$ .

[2]

Determine the values of $a$ and $b$ .

[3]

Find the equation of the normal to the graph of $f$ at $x=a$ .

[3]

Find the equation of the tangent to the graph of $f$ at $x=b$ .

[2]

Find the coordinates ( $p, q$ ) of the intersection point $P$ .

[3]

Sketch the graph of $y=f^{\prime}(x)$ , indicating the x-intercepts.

[3]

Question 9

SLPaper 1

The function $f$ is defined for all $x \in \mathbb{R}$ . The line with equation $y=8 x-7$ is the tangent to the graph of $f$ at $x=2$ .

The function $g$ is defined for all $x \in \mathbb{R}$ where $g(x)=x^{2}+x$ , and $h(x)=f(g(x))$ .

Write down the value of $f^{\prime}(2)$ .

[1]

Find $f(2)$ .

[1]

Find $h(2)$ .

[2]

Hence, find the equation of the tangent to the graph of $h$ at $x=2$ .

[3]

Question 10

SLPaper 1

Consider the function $f(x)=x^{3}-3 x^{2}+k x+2$ , where $k$ is a constant. The line $y=m x+c$ is the tangent to the graph of $y=f(x)$ at the point where $x=1$ . The function $g(x)=x^{2}-4 x$ is defined for all $x \in \mathbb{R}$ , and $h(x)=f(g(x))$ .

Find $f^{\prime}(x)$ .

[2]

Given that the gradient of the tangent at $x=1$ is 3 , find the value of $k$ .

[3]

Find the equation of the tangent to the graph of $y=f(x)$ at $x=1$ .

[2]

Find $h^{\prime}(x)$ using the chain rule.

[3]

Determine the gradient of the normal to the graph of $y=h(x)$ at $x=2$ .

[3]

SL 5.4—Tangents and normal Questionbank