How do we calculate the gradient of a curve or the area underneath it?

Key questions

  1. 1

    How can we find the derivative or integral of a power of \(x\)?

  2. 2

    How are differentiation and integration related?

  3. 3

    How can we locate a maximum or minimum point on a graph?

  4. 4

    How can we find and interpret the tangent or normal to a curve at a point?

  5. 5

    How and when can we use integration to find the area under a curve?

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Resource type Title
Building blocks Gradient spotting
Investigation Binomials are the answer!


Resource type Title
Fluency exercise Floppy hair
Fluency exercise Integral chasing
Package of problems Tangent or normal
Food for thought Stretch the function
Investigation Meaningful areas
Bigger picture Calculus of variations—finding the best possible way
Bigger picture Queen Dido's problem

Review questions

Title Ref
A cubic has one real root—can we find an approximation to it? R8231
Can we explain why this integral is zero? R7724
Can we find a curve from its gradient function? R5768
Can we find a percentage increase using the gradient? R6345
Can we find the area between a parabola and a line? R9357
Can we find the area enclosed by this quadratic and the $x$-axis? R9138
Can we find the area inside a parabola, a tangent, and the $x$-axis? R6584
Can we find the area inside a straight line, a parabola and the x-axis? R7444
Can we find the area of the triangle created by a common tangent? R6877
Can we find the area under a parabola? R6067
Can we find the centre of mass of a solid hemisphere? R5584
Can we find the locus of a midpoint created by a parabola? R8399
Can we find the triangle areas given by a tangent? R8286
Can we show these two cubic curves touch? R8543
Can we sketch the graph of $y^2 = x^3$? R6372
Can we track this constantly growing patch of fluid? R6959
Given the line $PRQ$, when is $OP + OQ$ a minimum? R7586
Given three clues, can we find the equation of the curve? R7734
How many common tangents can two parabolas have? R7750
If we increase $x$ by a small amount, what happens to $y$? R9942
If we take a slice from this sphere, where is its centre of gravity? R7628
Is the gradient function increasing or decreasing on this curve? R7433
What (approximately) is the square root of $26$? R8065
What can we say about the combined area of these two shapes? R9599
What happens if we expand, then differentiate? R8931
What happens if we integrate, then differentiate? R8716
What is the gradient of this curve at its $x$-intercepts? R7391
What volume is generated when $y = ax - x^2$ is rotated about the x-axis? R5007
What's the least surface area of this rocket-shaped solid? R8548
What's the most economical speed for this ship's journey? R7280
When does $y=kx^3-(k+1)x^2+(2-k)x-k$ have a minimum at $x =1$? R8441
When does the normal to a parabola pass through this point? R6352
When does this definite integral have a minimum? R7236
When is the area of a face of this pyramid a minimum? R6820
When is the area of this region bigger than the area of this one? R8350
When is the volume of this cuboid a maximum? R9260
When is the volume of this cylinder inside a cone a maximum? R7785
Where do two tangents cross, and where do two normals cross? R5703
Where does $f(x) = x^2 - 2px + 3$ have its minimum? R5732