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L3 Differentiation

Question Bank

Basic Derivatives Gradients & Rates of Change Limits & Continuity Tangents & Normals Stationary & Inflection Points Implicit Differentiation Parametric Related Rates Optimisation Kinematics Verifying Differential Equations
Achieved

Find the $x$-coordinate of the point on the curve $y = 6x + e^{-2x}$ where the tangent is parallel to the $x$-axis.

Give your answer in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Achieved

Find the $x$-coordinates of the stationary points of the curve

$ y = \frac{x^2 + 16}{2x} $

You must use calculus and show any derivatives that you need to find when solving this problem.

Achieved

Find the $x$-coordinates of the stationary points of the curve $y = x^2e^{-x/2}$.

You must use calculus and show any derivatives that you need to find when solving this problem.

Achieved

Find the $x$-coordinate of the stationary point of the curve

$ y = \frac{3 - \ln x}{2x}, \quad x > 0 $

Give your answer in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Achieved

Find the $x$-coordinates of the stationary points of

$ f(x) = \frac{x^2 + 5}{x + 2} $

You must use calculus and show any derivatives that you need to find when solving this problem.

Achieved

Find the exact $x$-coordinates of the stationary points of the curve $y = x - 2\ln(x^2 + 1)$.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

The curve $y = x\sqrt{12 - x}$, $x \leq 12$, has one stationary point.

Find its coordinates.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the stationary point of the curve $y = x^2 + \dfrac{16}{x}$, and determine its nature.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

For what values of $x$ is the function $f(x) = x^2 - 10x + 12\ln x$, $x > 0$, decreasing?

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the stationary point of the curve $y = (3 - x)e^{x/2}$, and determine its nature.

Give the coordinates in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the points of inflection on the curve $y = \dfrac{4}{x^2 + 3}$.

You can assume that your points found are actually points of inflection.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the point of inflection on the curve

$ y = \frac{1 + \ln x}{x}, \quad x > 0 $

You can assume that your point found is actually a point of inflection.

Give your answer in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

For what values of $x$ is the function $f(x) = (x^2 - 2x - 1)e^x$ decreasing?

Give your answer in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the stationary points of the curve $y = (x^2 - 3)e^{-x}$, and determine the nature of each.

Give the coordinates in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

The function $f$ is defined for $x \neq 2$ by

$ f(x) = 4x + 1 + \frac{9}{x - 2} $

For what values of $x$ is $f$ increasing?

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the stationary points of the curve

$ y = \frac{x^2 + 3}{x - 1} $

and determine the nature of each.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the point of inflection on the curve $y = x^3(2\ln x - 1)$, $x > 0$.

You can assume that your point found is actually a point of inflection.

Give your answer in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Consider the family of curves

$ y = \ln x - kx^2, \quad x > 0 $

where $k$ is a positive constant.

Show that each curve has exactly one turning point, and that it is a maximum.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

For what values of $x$ is the function $f(x) = \dfrac{x + 1}{x^2 + 8}$ increasing?

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of all the points on the curve

$ y = 2\sin x + \cos 2x $

for $0 < x < 2\pi$, where the tangent is parallel to the $x$-axis.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the stationary points of the curve

$ y = \frac{(\ln x)^2}{x}, \quad x > 0 $

and determine the nature of each. Give the coordinates in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of all the points of inflection on the curve $y = xe^{-x^2}$.

You can assume that your points found are actually points of inflection.

Give your answers in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

For what values of $x$ is the function $f(x) = 3\ln(x^2 + 1) - 2x$ increasing?

Give your answer in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the stationary points of the curve

$ y = \frac{\sin x}{2 + \cos x}, \quad 0 \leq x \leq 2\pi $

and determine the nature of each. Give the coordinates in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

Find the coordinates of the stationary points of the curve $y = x^3e^{-x}$, and determine the nature of each.

Give the coordinates in exact form.

You must use calculus and show any derivatives that you need to find when solving this problem.

Excellence

The curve

$ y = \frac{ax + b}{x^2 + 3}, $

where $a$ and $b$ are constants, has a turning point at $(1, 2)$.

Find the values of $a$ and $b$, and show that this turning point is a maximum.

You must use calculus and show any derivatives that you need to find when solving this problem.

Excellence

The curve $y = axe^{bx}$, where $a$ and $b$ are constants, has a point of inflection at $\left(4, \dfrac{8}{e^2}\right)$.

Find the values of $a$ and $b$. Hence find the exact coordinates of the turning point of the curve, and determine its nature.

You must use calculus and show any derivatives that you need to find when solving this problem.

Excellence

Consider the function $f(x) = (x^2 + k)e^x$, where $k$ is a constant.

(a) Find the values of $k$ for which the graph of $y = f(x)$ has two turning points.

(b) Find the value of $k$ for which the graph has exactly one stationary point, and show that this stationary point is not a turning point.

You must use calculus and show any derivatives that you need to find when solving this problem.

Excellence

The curve $y = \ln(x^2 + k)$, where $k$ is a positive constant, has two points of inflection.

(a) Find the coordinates of the points of inflection, in terms of $k$.

(b) Find the exact value of $k$ for which the tangents to the curve at the points of inflection pass through the origin.

You must use calculus and show any derivatives that you need to find when solving this problem.