# Solving Exponential Growth And Decay Problems

And, in case you were wondering where you get to take a derivative here, the answer is that you don't really: all the ‘calculus work’ was done at the point where we granted ourselves that all solutions to that differential equation are given in the form \$f(t)=ce^\$.

And, in case you were wondering where you get to take a derivative here, the answer is that you don't really: all the ‘calculus work’ was done at the point where we granted ourselves that all solutions to that differential equation are given in the form \$f(t)=ce^\$.First to look at some general ideas about determining the constants before getting embroiled in story problems: One simple observation is that \$\$c=f(0)\$\$ that is, that the constant \$c\$ is the value of the function at time \$t=0\$.

Such a relation between an unknown function and its derivative (or derivatives) is what is called a differential equation.

Many basic ‘physical principles’ can be written in such terms, using ‘time’ \$t\$ as the independent variable.

The main focus is on solving problems involving exponential growth and decay. 'Exponential graphs' is a 'Core Maths' resource that asks students to match graphs into different classes.

'Logarithms' is a RISP 'always, sometimes or never true' activity where students need to be familiar with logs in different number bases.

The general idea is that, instead of solving equations to find unknown numbers, we might solve equations to find unknown functions.

There are many possibilities for what this might mean, but one is that we have an unknown function \$y\$ of \$x\$ and are given that \$y\$ and its derivative \$y'\$ (with respect to \$x\$) satisfy a relation \$\$y'=ky\$\$ where \$k\$ is some constant.

The next step is to find the trend by noting that we are left with a certain percentage of the substance.

One you have the trend you will use this to calculate the amount of substance left in said hours, in this case 6 hours.

The interactive file can be used to demonstrate some of the important aspects of growth and decline, offering work on geometric progressions.

The main focus is on solving problems involving exponential growth and decay.

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