Before continuing with your study of differential calculus, it's worth pausing and reviewing the important ideas for this topic.
When we refer to the average rate of change of a function over a given domain, we are referring to the gradient between the two points on the curve, hence providing us with an average change.
When we refer to the instantaneous rate of change of a function we want to know the rate of change at a particular instant, or point, on the curve.
We can measure this by drawing a tangent to the curve at our point of focus and calculating its gradient to determine the instantaneous rate of change. We can also determine the equation of our tangent line if we wish.
Alternatively, we can use the limiting chord process and examine the average rate of change over increasingly smaller intervals of the domain and determine the value of the limit, or gradient, that we approach.
Be sure to review the lessons linked above to consolidate each of these important ideas and then complete the questions in this exercise.
The value of a particular coin can be modelled by the equation $y=80\left(2^x\right)$y=80(2x), where $x$x is the number of years from now.
What is the average rate of change of its value over the interval $\left[0,4\right]$[0,4]?
What is the gradient of the tangent at the given point?
Consider the curve $f\left(x\right)$f(x) drawn below along with $g\left(x\right)$g(x), which is a tangent to the curve.
What are the coordinates of the point at which $g\left(x\right)$g(x) is a tangent to the curve $f\left(x\right)$f(x)?
Note that this point has integer coordinates. Give your answer in the form $\left(a,b\right)$(a,b).
What is the gradient of the tangent line?
Hence determine the equation of the line $y=g\left(x\right)$y=g(x).
Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods
Apply differentiation methods in solving problems