Unit Resources:
- Daily Plan
- Lesson Package
- Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989: Chapter 10, Chapter 11
Homework:
HW1: Original and Answer Key
HW2: Original and Answer Key
Area HW: Original and Answer Key
Old Exam: Original and Answer Key
Antiderivatives
If we know the derivative of the function, we can use reverse thinking to find the 'antiderivative' and write a general expression for the function.
- Polynomial functions - increase by a degree and divide by the increased degree.
- Trig functions - use knowledge of derivatives to find antiderivatives, and divide by the derivative of 'theta'.
- Exponential functions, base e - use our knowledge of 'repeat the function' and as with other functions, we will divide by the derivative of the exponent.
- Logarithmic functions - if we are working through a polynomial function, and we increase from a degree of -1 to zero... this would result in divide by zero too... so we use our knowledge of derivatives of "lnx" to work backwards in this situation.
Part 1 - determine the antiderivative; not solving for the value of the constant. Part 2 - using information about the original function to solve for 'c' before determining the equation of the function.
Basic Integration
The rules of integration are the same as antiderivatives... polynomials increase by a degree and divide by the new larger degree; exponential functions have a repeat function part and a divide by the derivative of the exponent part; trigonometry ratios and logarithmic functions have their own pattern of working backwards, but both involve divide by a derivative.
Chain Rule:
- When taking derivatives, part of the process is to multiply by the derivative of the function.
- When working backwards, part of the process will be to divided by the derivative of the function.
Area Under a Curve - Evaluate definite integrals.
- We can find the area under a curve by taking the sum of an infinite number of trapezoids between two x-values... which we calculate using the integral of the curve and evaluating: F(x-upper value) - F(x-lower value).
- We do not need to worry about the constant term from a indefinite integral when calculating a definite integral as the value of 'c' becomes non-existent after subtracting.
- Notes
Integration By Parts (reverse product rule).
- To reverse the product rule we need to decide which part of the integral is the 'function' and which part is the 'derivative'.
- Chose the function and derivative so the new integral is 'less complicated' than the original.
- Notes