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These are the Transum resources related to the statement: "Understand and use the derivative of f (x) as the gradient of the tangent to the graph of y = f ( x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change, sketching the gradient function for a given curve, second derivatives, differentiation from first principles for small positive integer powers of x and for sin x and cos x. Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection".

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

• Differentiation Video A reminder of how to differentiate different types of functions and how to find the equations of tangents and normals.
• Differentiation Practise the technique of differentiating polynomials with this self marking exercise.

Here are some exam-style questions on this statement:

• "A function is given as \(f(x)=3x^2-6x+4+\frac3x,-2\le x \le 4, x\ne 0\)." ... more
• "Consider the graph of the function \(f(x)=7-3x^2-x^3\)" ... more
• "A child's play tent is made in the shape of half a cylinder. It is constructed from a fibreglass frame with material pulled tightly around it. The fibreglass frame consists of a rectangular base, two semi-circular ends and two further support rods, as shown in the following diagram." ... more
• "A package is in the shape of a cuboid and has a length \(l\) cm, width \(w\) cm and height of 12 cm." ... more
• "Consider the function \(f(x)=6 - ax+\frac 3{x^2},x\neq 0\)" ... more
• "If \(f(x)=xsinx\), for \(-3\le x\le3\)" ... more
• "Consider the cubic function \(f(x)=\frac{1}{6}x^3-2x^2+6x-2\)" ... more
• "Consider the function \(f(x)=x^3-9x+2\)." ... more
• "Let \(f(x)=\frac{g(x)}{h(x)}\), where \(g(3)=36\), \(h(3)=12\), \(g'(3)=10\) and \(h'(3)=4\). Find the equation of the normal to the graph of \(f\) at \(x=3\)." ... more
• "The function \(f\) is defined as follows:" ... more
• "Consider the functions" ... more
• "Let \(f(x)=jx^3+jx^2+kx+m\) where \(j, k\) and \(m\) are constants." ... more
• "The following diagram shows part of the graph of \(y=f (x)\)" ... more
• "Let \(f(x) = \frac{ln3x}{kx} \) where \( x \gt 0\) and \( k \in \mathbf Q^+ \)." ... more
• "Consider the function \(f(x)=\frac{20}{x^2}+kx\) where \(k\) is a constant and \(x\neq0\)." ... more

See all these questions

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

• Calculus It is said that the word calculus comes from the Latin word for the small pebble used for counting and calculations. The two major branches, differentiation and integration, are studied by pupils only towards the end of their school days but does then form a major part of their studies. A course in calculus is a prerequisite for other, more advanced courses in mathematical analysis.