Calculus meets Functions

Functions
Calculus

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How does calculus deepen our understanding of functions, and vice versa?

Key questions

1
How does calculus help us to sketch graphs of functions?
2
How can we classify stationary points on graphs of functions?
3
How does the language of functions help us to talk about calculus, and vice versa?
4
What does the second derivative tell us about a function?
5
How do transformations of functions affect derivatives and integrals?

Station guide

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Introducing...

Resource type	Title
Building blocks	Gradients of gradients

Developing...

Resource type	Title
Building blocks	Thinking constantly
Package of problems	Additional integrals
Many ways problem	Can you find... curvy cubics edition
Many ways problem	Two-way calculus
Problem requiring decisions	Choose your families
Problem requiring decisions	Keep your distance
Food for thought	Slippery slopes
Food for thought	What else do you know?
Investigation	Curvy cubics
Go and think about it...	Average turning points
Resource in action	Two-way calculus - teacher support

Review questions

Title	Ref
Can we maximise the volume of this metal tin?	R6180
Can we show that the area of this hexagon is $2(x^2 + xy + y^2)$?	R8987
Can we sketch the graph of this piecewise function?	R7196
Can we solve these simultaneous equations with integrals?	R6076
For the functions $f$ and $g$, is $g(f(A))$ always bigger than $f(g(A))$?	R9099
Given $f'(x)$ and $g'(x)$, what is $f''(2x)$?	R8588
How many stationary points does $y = 2x^3 - 6x^2 + 5x - 7$ have?	R9582
How small can this triangle be?	R7720
If $f(x)$ always satisfies this equation, what is $\int_{-1}^1 f(x)\: dx$?	R8105
If $y=f(x)$ has a turning point when $x=\frac{1}{4}$, can we find $\lambda$?	R7998
If the trapezium rule overestimates $\int^1_0 f(x)\, dx$, what can we deduce?	R5324
If we increase $r$ by a small amount, what happens to $A$?	R5329
If we know $f(x)$, can we find $\int_{-1}^1 f(x^2-1) \:dx$?	R5549
What can we say if $y = m(x-a)$ is tangent to $y = x^3 - x$?	R8287
What does it mean if a function $f(x)=f(2\alpha-x)$?	R5090
What's the area between $f_k(x) = x(x-k)(x-2)$ and the $x$-axis?	R8130
What's the area enclosed by $y=x(x-3)^2$ and a tangent?	R7386
What's the minimum area of this variable triangle?	R7212
When does $3x^4 - 16x^3 +18x^2 + k = 0$ have four real solutions?	R8959
When does $3x^4-16x^3+18x^2=k$ have exactly two real roots?	R6537
When does a cubic curve have two turning points?	R7861
When does this cubic equation have distinct real positive solutions?	R9781
When does this cubic equation have three distinct real roots?	R6703
When does this function of two variables have a minimum?	R9671
When is $I(c) = \int^1_0 2^{-(x-c)^2} dx$ largest?	R7461
When is $f(x) = 2x^3 − 9x^2 + 12x + 3$ smallest over this range?	R6868
When is the cost of producing this chemical a minimum?	R6867
When is this integral less than or equal to $1 - A/(n+B)$?	R8396