Applications of Differentiation Practice Questions

The equation of a curve is $y = x^3 - 4x^2 + 7x$

Show that the curve is increasing at the point where $x = 3$

Work out $\dfrac{d^2y}{dx^2}$

Work out the value of $\dfrac{d^2y}{dx^2}$ when $x = -1$ and state whether the gradient of the curve is increasing or decreasing at this point.

The equation of a curve is $y = x^3 - 4x^2 + 6x - 1$

Work out the equation of the normal to the curve at the point $(2, 3)$

Give your answer in the form $y = mx + c$

A curve has equation $y = x^3 - 3x^2 - 9x + 2$

Work out the coordinates of the two points on the curve where the tangent is parallel to the line $y = 15x - 4$

The equation of a curve is $y = 2x^3 + 3x^2 - 12x + 1$

Work out the range of values of $x$ for which $y$ is a decreasing function.

The equation of a curve is $y = 2x^3 - 9x^2 + 12x - 3$

Work out the stationary points of the curve and determine their nature.

You must show your working.

Tick one box for each statement.

Work out the values of $a$ and $b$.

Determine the nature of the stationary point where $x = 1$

The function $f$ is given by $f(x) = x^3 + kx^2 + 12x + 7$ where $k$ is a constant.

$f$ is an increasing function for all values of $x$.

Work out the possible values of $k$.

Give your answer as an inequality.

Show that the volume, $V$ cm$^3$, of the box is given by

$$V = 150x - \tfrac{1}{2}x^3$$

Use calculus to work out the maximum value of $V$.

You must show that your value is a maximum.

Show that the tangent to the curve at $P$ has equation $y = 4x - 14$

The tangent at $P$ crosses the $x$-axis at $A$.

The normal to the curve at $P$ crosses the $x$-axis at $B$.

Work out the area of triangle $APB$.