Understanding the meaning of integrals in application.

Study how the general expression for the natural log changes the shape of the graph.

Applications of integration involving area between curves. Integral of top curve minus integral of bottom curve to get difference. When to consider the negative of the integral for the purpose of area and thinking about the difference between the total area and the value of an integral.

Remember to use sum of integrals to increase /decrease the domain of an integral over a function. I suppose that when dealing with the type of problem we looked at today, it’s good to think about it geometrically (I.e., consider how each integral is associated with each region and how those regions interact) as well as in terms of manipulation of integrals (I.e., using rules for operations on integrals). In terms of looking at upper and lower sums for approximations of integral, just remember that for each rectangle, you have a set domain along the x-axis and you change the height of the rectangle by considering which value of x, in this domain, to evaluate f(x) at. For the upper estimate, pick values of x such that the height of each rectangle is maximised and for the lower estimate, pick the value of x in each interval such that each rectangle is minimised in height.

Relationship between integration and area under and between curves. Solving equations/systems of equations.

Formulation of visual and worded problems into a system of mathematical objects that we can use and manipulate i.e. functions and equations. This often involves identifying variables/parameters and working out how they relate to each other. Also need to ensure you understand exactly what the the question is asking for because this informs the decisions you make in setting up your mathematical system. I understand this takes time and is therefore difficult under exam conditions. Just keep practicing at these sorts of problems. If you can’t get them out under exam conditions, save them for later and think them through when you have more revision time.

The sign diagram of f'(x) tells us information about f'(x) which we can use to draw a sketch of f(x) because we will know when f(x) is increasing, decreasing, and stationary. We can also learn about the concavity of f(x) by looking at f'(x): when f'(x) is increasing, f(x) is concave up (convex); when f'(x) is decreasing f'(x) is concave down. Or we can use the second derivative: when f''(x) is positive, f(x) is concave up; when f''(x) is negative, f(x) is concave down.

Some things to remember: f'(x) = 0 -> stationary point, stationary point with change is sign of f'(x) -> turning point, stationary point with no change in sign -> stationary inflection point. f''(x) = 0 and change in sign of f''(x) -> inflection point (change in curvature). If you have vertical asymptotes in f(x), then you will have the same vertical asymptotes in f'(x) and f''(x) and f'''(x),... but the behaviour of the each graph near these asymptotes will vary (it will sort of alternate). However, just because f'(x) has an asymptote do not mean f(x) must have and asymptote.