Consider the function f defined by $f(x)=5 x-\tan x, 0 \leq x<\frac{\pi}{2}$.

a. Find the equation of the tangent to the graph of $f$ at $x=\frac{\pi}{4}$.

b. There is a point A on the graph of f where the normal to the graph is vertical. Determine the coordinates of A.

The motion of an object moving along a horizontal line with displacement $s(t)=2 t^3-21 t^2+60 t+3$, for $0 \leq t \leq 7$ ( s in metres, t in seconds), can be modelled by the following diagram.

The object is moving along line s. The purple curve above the line is used to show the direction the particle is moving and its position at key times.

Use your GDC to find $\int_0^7|v(t)| \mathrm{d} t$ and compare it to your answer in the previous question.

What does the cosine function tell you about the graph of the sine function for any given value in the domain of the sine function?

Sketch a graph of $f(x)=\cos x$, for $-2 \pi \leq x \leq 2 \pi$ and repeat the process you followed in the previous questions. Hence, make a suggestion about the function which is the derivative of the $f(x)=\cos x$.

Find the derivative of the function below.

$$y=\cos \frac{\pi}{x}$$

A portion of the graph of $f(x)=\mathrm{e}^x \sin x$ is shown in the diagram. The graph has x-intercepts at x=0 and x=k.

a. Find the value of k.

b. Find the area of the shaded region.

Find the indefinite integrals in the question below.

$$\int(2 x-1) \cos \left(2 x^2-2 x\right) \mathrm{d} x$$

Find the derivative of the function below.

$$y=\cos x+x \sin x$$

Find the indefinite integral.

$$\int \cos \left(\frac{x}{2}\right) \mathrm{d} x$$

Consider the function g defined by $g(x)=\sin ^2 x-5 x$

a. Find an expression for $g^{\prime}(x)$.

b. Hence, show that g is decreasing on all its domain.

It is given that $y=f(x), \frac{f(b)-f(a)}{b-a}$ is the average rate of change of $f$ over $[a, b]$. It is also possible to represent the instantaneous rate of change of a function, which is the rate of change at a certain point.

What represents the instantaneous rate of change of a function at a given point?

The velocity, $v \mathrm{~m} \mathrm{~s}^{-1}$, of an object moving along a straight line at time t seconds is given by $v(t)=\mathrm{e}^{3 \cos t}-2$, where $t$ is measured in seconds, for $0 \leq t \leq 6$.

a. Find the acceleration of the object at t=1.

b. Determine the value(s) of t when the particle has a velocity of $8 \mathrm{~m} \mathrm{~s}^{-1}$.

c. Find the displacement of the particle after 6 seconds.

d. Find the distance travelled by the particle during the time interval $0 \leq t \leq 6$.

Rewrite each definite integral using a $u$-substitution. Then find the exact value of the definite integral without using a GDC. Check your answer with a GDC.

$$\int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{3}} \mathrm{e}^x \sin \left(\mathrm{e}^x\right) \mathrm{d} x$$

What is the derivative of f(x) = sin x?

Water is dripping into a container over a ten-hour period. The height of the water in the container increases at a rate of $r(t)=7 t^2 \mathrm{e}^{-1.2 t}$ centimetres per hour, where t is time in hours. If the height of the water in the container was 4.5 cm at the beginning of the ten-hour period, what was the height of the water after the ten hours?