Level 2 Calculus

(a)

Find the gradient of the function $f(x) = (2x + 3)^2$ at the point where $x = 1$.

(b)

Shown below is the graph of a function $y=f(x)$. Sketch the graph of the derivative $y=f'(x)$.

$y = f(x)$

(c)

The graph of the function $f(x) = x^3 - 6x^2 + 5$ has a stationary point at $x=0$. Use calculus to verify that this point is a local maximum.

(d)

The equation of a curve is $y = x^3 - 3x^2 + 5$. Find the equation of the tangent to the curve at the stationary point where $x > 0$.

(e)

Toblerone is investigating changing their chocolate design. Their current triangular prism box uses $270 \text{ cm}^2$ of cardboard and contains a volume of $148.5 \text{ cm}^3$.

They are looking to change their box to a square-based prism (cuboid) that uses the same amount of cardboard, $270 \text{ cm}^2$, but is designed to maximise volume.

Use calculus to determine how much extra volume they could hold compared to their current box (which is $148.5 \text{ cm}^3$).

(a)

The formula for the area of a square is $A = s^2$, where $s$ is the side length. Find the rate at which the area is changing with respect to its side length at the instant when the side is 10 cm.

(b)

The derivative of a function is $\frac{dy}{dx} = 3x^2 - 12x + 9$. If the curve has a stationary point at $(1, 5)$, find the equation of the curve.

(c)

A small business produces and sells wooden craft signs. The total daily profit, $P$ (in dollars), depends on the number of signs produced, $n$, and is modelled by the function:

$$ P(n) = -0.01n^2 + 1.2n - 22 $$

Use calculus to find the maximum possible daily profit.

(d)

(i) A model rocket launches vertically from rest at ground level. Its acceleration for the first stage is modelled by the function $a(t) = 0.3t + 2 \text{ m/s}^2$. The engine cuts out exactly when the rocket's velocity reaches $v = 35 \text{ m/s}$. Find the time, $t$, at which the engine cuts out.

(ii) At $t=10$ seconds, the main engine cuts out and a secondary engine provides a constant acceleration of $4.5 \text{ m/s}^2$ for a further $15$ seconds. Calculate the total height of the rocket above the ground after it has been travelling for a total of $25$ seconds.

(e)

The line $y = 3x + k$ is tangent to the graph of $y = x^4 - 18x^2 + 3x + 1$ at two distinct places. Find the required value of $k$ and verify that the line is tangent at the two intersection points.

(a)

Find the gradient of the tangent to the curve $f(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 4x + 1$ at the point where $x = -2$.

(b)

The total volume, $V$ (in $\text{cm}^3$), of water in a tank $t$ seconds after a valve is opened is modelled by the function:

$$ V(t) = 5000 + 150t + 5t^2 $$

How many seconds will it take for the rate of change of the volume of water to be $250 \text{ cm}^3\text{/s}$?

(c)

The graph of a function $f(x) = ax^3 - 9x^2 + 5$ has a turning point when $x = 3$. Find the value of the constant $a$.

(d)

For what values of $x$ is the function $f(x) = x^3 - 9x^2 + 15x + 4$ an increasing function?

(e)

Shown is the graph of $y=f'(x)$. Sketch a possible graph for $y=f(x)$.

(f)

In calculus a fundamental theorem is the Mean Value Theorem (MVT). In simple terms the MVT says that if you draw a line connecting two points on a smooth curve, there must be at least one place in between where the curve's actual gradient (the instantaneous gradient) is exactly the same as the average gradient.

The function shown in the diagram is $f(x) = \frac{k}{2}x^3 + kx^2 - kx + 1$, where $k$ is a constant. The secant line connects the points $(-2, 2k+1)$ and $(2, 6k+1)$.

Find the exact $x$-coordinates of all points in the interval $-2 < x < 2$ where the instantaneous gradient is equal to the average gradient between the two given points.