### Introduction

This STEP support module derives some more differentiation results and asks you to sketch some graphs. Previous assignments can be found here, but you can do this one without having done the others first.

STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").

### About this assignment

The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.

The **warm up** for this assignment derives the product rule for differentiation, and asks you to prove some things about the exponential function from its definition as an infinite series.

The **main STEP question (2015 STEP 1 Question 1)** is about sketching graphs and using your sketches to work out how many solutions there are to a particular equation.

The **final question** is about Euler's formula for convex polyhedra.

For more on Euler's formula, see this Plus article.

This Nrich problem leads you through a proof of Euler's formula.

See this Advanced Problem solving module for some more on some calculus techniques.

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 22”** file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.

Here is a Worked Video Solution to the STEP question from this assignment.