(a)

Find $\int(6 x+7) \mathrm{d} x$ .

[ 3 ]

(b)

Given $f^{\prime}(x)=6 x+7$ and f(1.2)=7.32, find f(x).

[ 3 ]

[Maximum number: 7]

Consider the function defined by $f(x)=x^{2}-8 x$ . The graph of f passes through the point A(3,-15).

(a)

(i)

Find the gradient of the tangent to the graph of f at the point A .

[ 3 ]

(b)

Write down the equation of the normal to the graph of f at point A .

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

[ 1 ]

(c)

Find the coordinates of B.

[ 3 ]

[Maximum number: 4]

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

(a)

Find $f^{\prime}(1)$ .

[ 2 ]

(b)

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

[ 2 ]

[Maximum number: 5]

Let $f(x)=\ln x-5 x$ , for x>0.

(a)

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

[ 2 ]

(b)

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

[ 1 ]

(c)

Solve $f^{\prime}(x)=f^{\prime \prime}(x)$ .

[ 2 ]

[Maximum number: 3]

Consider the function $f(x)=\frac{(x-1)^{2}}{x}$ , where $x \in \mathbb{R}, x \neq 0$ .

(a)

Hence, find $\int f(x) \mathrm{d} x$ .

[ 3 ]

[Maximum number: 2]

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

(a)

Find the coordinates of A.

[ 2 ]

[Maximum number: 2]

Let $f(x)=\frac{6 x^{2}-4}{\mathrm{e}^{x}}$ , for $0 \leq x \leq 7$ .

(a)

The graph of f has a maximum at the point A . Write down the coordinates of A .

[ 2 ]

[Maximum number: 5]

The derivative of a function g is given by $g^{\prime}(x)=\cos x+\mathrm{e}^{2 x}$ , where $x \in \mathbb{R}$ .
Given that g(0)=7, find g(x).

(a)

Hence, find the value of $\int_{1}^{9}\left(\frac{3 \sqrt{x}-5}{\sqrt{x}}\right) \mathrm{d} x$ .

[ 4 ]

[Maximum number: 6]

Let $f(x)=6 x^{2}-3 x$ . The graph of f is shown in the following diagram.

(a)

Find $\int\left(6 x^{2}-3 x\right) \mathrm{d} x$ .

[ 2 ]

(b)

Find the area of the region enclosed by the graph of f, the x-axis and the lines x=1 and x=2.

[ 4 ]

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