If we draw in the tangents to the curve, you will notice that if the gradient of the tangent is positive, then the function is increasing and if the gradient is negative then the function is ...

The tangent will have an equation in the form \(y = mx + c\) so to find the equation you need to find the values of \(m\) and \(c\). First, find \(m\), the gradient of the tangent. On a diagram ...

Find \(\ds \lim_{h\to 0}\frac{f(1+h)-f(1)}{h}\) where \(\ds f(x)=\frac{3x+1}{x-2}\text{.}\) What does the result in (a) tell you about the tangent line to the graph ...

Express the polar equation \(r=\cos 2\theta\) in rectangular coordinates. Find the maximum height above the \(x\)-axis of the cardioid \(r=1+\cos \theta\text{.}\) Sketch the graph of the curve whose ...

From trigonometry you may dimly remember the tan, or tangent, function, which translates between an angle and its associated ...

In place of such an intermediate, Mitchell proposed a proton gradient across a membrane: the proton motive force (Mitchell 1961). It works much like a hydroelectric dam. The energy released by the ...

What have we learned? For some, this progression is proof of a ‘slippery slope’ – once one element is allowed, others will inevitably follow until almost anything is permitted. For others ...

A core task in single-cell data analysis is recovering the latent dimensions encoding the genetic and epigenetic landscapes inhabited by cell types and lineages. However, consensus is lacking for ...

The Ring of Fire is home to 75% of the world's volcanoes and 90% of its earthquakes. 2 min read The Ring of Fire is a roughly 25,000-mile chain of volcanoes and seismically active sites that ...

We just admitted our largest and most diverse class in our history. Applying to UC is a lot simpler than you think. That’s because you can apply for as many campuses as you like — with just one ...

His firm is behind the design of a new artificial ski slope in Denmark's capital Copenhagen, built on the roof of a huge incinerator that burns waste to produce heat and electricity. "A power ...