Question 1

Using the definition of a derivative, find the derivative of $\frac{1}{(2x+1)}$

Accepted Answer

$\frac{-2}{(2x+1)^2}$

Answer

$\frac{-2}{(2x+1)}$

Answer

$\frac{-1}{(2x+1)^2}$

Answer

$\frac{-1}{(2x+1)}$

Question 2

Differentiate from first principles the function $f(x)=3x^3-x$.

Accepted Answer

$9x^2-1$

Answer

$9x^2-x$

Answer

$6x^2-1$

Answer

$9x^3-1$

Question 3

The slope of the tangent line for $f(x)=3x^3-x$. at $x=-1$ is

Accepted Answer

$8$

Answer

$-10$

Answer

$-8$

Answer

$10$

Question 4

The curve $y=-ax^3+7x^2-8x-5$ has a turning point at $x= -2$.  Determine the value of a.

Accepted Answer

$-3$

Answer

$3$

Answer

$\frac{5}{3}$

Answer

$-\frac{5}{3}$

Question 5

Find the smallest x-coordinate of all the points on the curve $y=2x^4+6x^3+\frac{7}{2} x^2-5x+\frac{3}{2}$ at which the tangent to the curve is parallel to the tangent at $(-1, 6)$.

Accepted Answer

$-1.5$

Answer

$-2.5$

Answer

$-1.8$

Answer

$-2.3$

Question 6

A camera at point $C$ is $3 m$ from the edge of a straight section of a road, as shown in the following diagram. The camera detects a car traveling along the road at $t = 0$. It then rotates, always pointing at the car, until the car passes $O$, the point on the edge of the road closest to the camera.
A car travels along the road at a speed of $24 ms^{−1}$. Let the position of the car be $X$ and let angle $OCX = θ$.
Find $\frac{dθ}{dt}$, the rate of rotation of the camera, in radians per second, at the instant the car passes the point $O$.

Accepted Answer

$8rad s^{−1}$

Answer

$6rad s^{−1}$

Answer

$9rad s^{−1}$

Answer

$7rad s^{−1}$

Question 7

The curve $C$ is defined by equation $xy-lny=1$, $y>0$.
Find $\frac{dy}{dx}$ in terms of $x$ and $y$.

Accepted Answer

$\frac{dy}{dx}=\frac{y^2}{1-xy}$

Answer

$\frac{dy}{dx}=-\frac{y^2}{1+xy}$

Answer

$\frac{dy}{dx}=\frac{y^2}{1+xy}$

Answer

$\frac{dy}{dx}=-\frac{y^2}{1-xy}$

Question 8

The curve $C$ is defined by equation $xy-lny=1$, $y>0$.
Find $\frac{dy}{dx}$ in terms of $x$ and $y$, then determine the equation of the tangent to $C$ at the point $(\frac{2}{e}, e)$

Accepted Answer

$y=-7.39x+8.15.$

Answer

$y=7.39x-8.15.$

Answer

$y=7.39x+8.15.$

Answer

$y=-7.39x-8.15.$

Question 9

Let $y=e^x sinx$.
Find an expression for $\frac{dy}{dx}$.

Accepted Answer

$\frac{dy}{dx}=e^x (sinx+cosx)$

Answer

$\frac{dy}{dx}=sinx+e^xcosx$

Answer

$\frac{dy}{dx}=e^x sinx+cosx$

Answer

$\frac{dy}{dx}=e^xcosx$

Question 10

Consider the curve with equation $x^3+y^3=4xy$.
Use implicit differentiation to find $\frac{dy}{dx}$

Accepted Answer

$\frac{dy}{dx}=\frac{4y-3x^2}{3y^2-4x}$

Answer

$\frac{dy}{dx}=\frac{4y-3x^2}{4y^2-3x}$

Answer

$\frac{dy}{dx}=\frac{3y-4x^2}{4y^2-3x}$

Answer

$\frac{dy}{dx}=\frac{3y-4x^2}{3y^2-4x}$

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