Calculus 1B

Net Area, Total Area and Area Between Curves

Distinguishing signed (net) area from total area and computing areas of regions bounded by curves and vertical lines. · 7 steps

Integration by Substitution

Reversing the chain rule by changing variables in indefinite and definite integrals. · 7 steps

Integration by Parts

Integrating products by transferring the derivative from one factor to the other. · 7 steps

Improper Integrals

Extending the definite integral to infinite intervals and unbounded integrands using limits. · 7 steps

Infinite Series, Geometric Series and Power Series

Convergence and summation of infinite series, with the geometric series as the master example. · 7 steps

Taylor and Maclaurin Polynomials and Series

Approximating a function near a point by the polynomial built from its derivatives there. · 7 steps

Differential Equations, Slope Fields and Separable Equations

What a differential equation and an initial value problem mean, and how to solve separable first-order equations. · 7 steps

First-Order Linear Differential Equations and Growth Models

Solving y' + P(x)y = Q(x) with an integrating factor, and reading qualitative behaviour from a model. · 7 steps

Second-Order Linear Equations with Constant Coefficients

Solving ay'' + by' + cy = f(x) via the characteristic equation plus a particular solution. · 7 steps

Polar Coordinates

Describing points, curves and regions in the plane by radius and angle instead of Cartesian coordinates. · 7 steps

Complex Numbers: Cartesian, Polar and Exponential Form

The three representations of a complex number and how arithmetic looks in each. · 7 steps

Complex Equations, Roots and Loci

Solving equations in C and describing sets of complex numbers geometrically. · 7 steps

Functions of Several Variables: Graphs, Level Curves and Limits

Extending the notion of a function to two variables and studying its surface, contours and limiting behaviour. · 7 steps

Partial Derivatives, Tangent Planes and Linearization

Differentiating a function of two variables one variable at a time and using it to approximate the surface by a plane. · 7 steps