Calculus: 112 IB Maths AA SL practice questions

(a) Find $\int (6x + 7)dx$[3]

(b) Given $f'(x) = 6x + 7$ and $f(1.2) = 7.32$, find $f(x)$[3]

Consider a function $f(x)$ such that $\int_1^6 f(x)\mathrm{d}x = 8$.

(a) Find $\int_1^6 2f(x)\mathrm{d}x$.[2]

(b) Find $\int_1^6 (f(x)+2)\mathrm{d}x$.[4]

Consider the function $f(x) = x^3 + 5x^2 - 8$, where $x \in \mathbb{R}$.

(a) Find $f'(1)$.[2]

(b) Find the equation of the tangent to the graph of $f$ at $x = 1$.[2]

Let $h(x) = \frac{6x}{\cos x}$. Find $h'(0)$.[6]

Consider the function $f(x) = x^2 + x + \frac{50}{x}$, $x \neq 0$.

(a) Find $f(1)$.[2]

(b) Solve $f(x) = 0$.[2]

The graph of $f$ has a local minimum at point A.

(c) Find the coordinates of A.[2]

Let $f'(x) = 6x^2 - 5$. Given that $f(2) = -3$, find $f(x)$.[6]

Consider $f(x) = x^2 \sin x$.

(a) Find $f'(x)$.[4]

(b) Find the gradient of the curve of $f$ at $x = \frac{\pi}{2}$.[3]

Let $f(x) = e^{-2x}$.

(a) Write down $f'(x)$, $f''(x)$, and $f^{(3)}(x)$.[3]

(b) Find an expression for $f^{(n)}(x)$.[3]

Given that $\frac{dy}{dx} = \cos\left(x - \frac{\pi}{4}\right)$ and $y = 2$ when $x = \frac{3\pi}{4}$, find $y$ in terms of $x$.[4]

The values of the functions $f$ and $g$ and their derivatives for $x = 1$ and $x = 8$ are shown in the following table.

Let $h(x) = f(x)g(x)$.

(a) Find $h(1)$.[2]

(b) Find $h'(8)$.[3]

(a) Find $\int(e^{4x} + 6) dx$.[3]

(b) It is given that $h'(x) = e^{4x} + 6$ and $h(1.5) = 105$. Find $h(x)$.[3]

(a) Find $\int (e^{3x} + 5) \, dx$[3]

It is given that $h'(x) = e^{3x} + 5$ and $h(2.1) = 185$.

(b) Find $h(x)$[3]

Consider the function defined by $f(x) = x^2 - 10x$. The graph of $f$ passes through the point $A(4, -24)$.

(a)(i) Find the gradient of the tangent to the graph of $f$ at the point $A$.[2]

(a)(ii) Hence, write down the gradient of the normal to the graph of $f$ at point $A$.[1]

(b) Write down the equation of the normal to the graph of $f$ at point $A$.[1]

(c) Find the coordinates of $B$.[3]

The normal to the graph of $f$ at point $A$ intersects the graph of $f$ again at a second point $B$.

Let $g(x) = \frac{\ln x}{x}$.

(a) Find $g'(x)$.[4]

(b) Find $\int g(x) \, dx$.[3]

The graph of a function $f$ passes through the point $(\ln 4, 20)$.

Given that $f'(x) = 6c^{2x}$, find $f(x)$.[7]