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Calculus 1

10 chapters71 lessons
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1

Limits and continuity

14 lessons

What a limit is, how to find one from a graph, a table or algebra, and what it means for a function to be continuous.

Review first: The difference quotient Infinite geometric series
1.1One sided limits and when a limit exists
1.2The limit laws
1.3Limits by algebra: factoring and rationalizing
1.4The squeeze theorem and the limit of sine x over x
1.5Continuity at a point
1.6Continuity on an interval
1.7Infinite limits and vertical asymptotes
1.8Discontinuities and how to remove them
1.9Making a piecewise function continuous
1.10Limits at infinity and horizontal asymptotes
1.11Comparing growth rates at infinity
1.12The intermediate value theorem
1.13The epsilon delta definition of a limit
1.14What a limit is: graphs and tablesretaught
2

The derivative

4 lessons

The derivative as the limit of a difference quotient, as a slope, as a function and as a rate of change.

2.1The derivative at a point
2.2Finding a derivative from the definition
2.3The derivative as a function
2.4Differentiability: corners, cusps and piecewise functions
3

Differentiation rules

8 lessons

The power, product and quotient rules and the derivatives of trigonometric and exponential functions.

3.1The power rule and basic differentiation rules
3.2The derivative as a rate of change
3.3Tangent lines and normal lines
3.4Derivatives of exponential functions
3.5The product rule
3.6The quotient rule
3.7Derivatives of the trigonometric functions
3.8Second and higher derivatives
4

Chain rule, implicit differentiation and inverse functions

6 lessons

Differentiating compositions, curves not given as functions, logarithms and inverse functions.

4.1The chain rule
4.2Implicit differentiation
4.3Derivatives of logarithms and logarithmic differentiation
4.4Derivatives of inverse functions
4.5Derivatives of inverse trigonometric functions
4.6Combining differentiation rules
5

Contextual applications: rates, related rates, linear approximation

4 lessons

Using derivatives to link changing quantities, approximate values and evaluate indeterminate limits.

5.1Related rates
5.2Linear approximation and differentials
5.3Newton's method
5.4L'Hopital's rule
6

Analyzing functions: extrema, the mean value theorem, curve sketching, optimization

10 lessons

What the first and second derivatives say about a graph and about motion, and how to find the best value.

Review first: Increasing, decreasing, constant, positive and negative intervals
6.1The mean value theorem
6.2The extreme value theorem
6.3Critical points and local versus absolute extrema
6.4Increasing and decreasing from the derivative
6.5Position, velocity and acceleration
6.6Extrema on a closed interval
6.7Concavity and inflection points
6.8The first and second derivative tests
6.9Curve sketching and connecting f, f' and f''
6.10Optimization
7

Riemann sums and the definite integral

5 lessons

Approximating accumulated change with sums of rectangles and trapezoids, and the definite integral as their limit.

Review first: Sigma notation
7.1Approximating area with Riemann sums
7.2The trapezoidal rule and Simpson's rule
7.3The definite integral as a limit of Riemann sums
7.4Signed area and integrals from geometry
7.5Properties of definite integrals
8

The fundamental theorem and antiderivatives

9 lessons

How differentiation and integration undo each other, and finding antiderivatives, including by substitution, long division and completing the square.

8.1The first fundamental theorem of calculus
8.2Graphing an accumulation function
8.3Antiderivatives and indefinite integrals
8.4The second fundamental theorem of calculus
8.5Antiderivatives of trig, exponential and log functions
8.6u-substitution
8.7Dividing before integrating
8.8Completing the square inside an integral
8.9Differential equations and their solutions
9

Applications of integration: area, volume, accumulation

7 lessons

Net change, distance traveled, area between curves, average value and volumes of solids.

9.1The net change theorem in context
9.2Position, displacement and distance from velocity
9.3Area between curves
9.4Average value of a function
9.5Volumes with known cross sections
9.6The disk method
9.7The washer method
10

Differential equations: slope fields and separation of variables

4 lessons

Writing differential equations from a description, picturing solutions with slope fields and solving separable equations, including exponential growth and decay.

Review first: Differential equations and their solutions
10.1Writing a differential equation from a description
10.2Slope fields
10.3Separable differential equations
10.4Exponential growth and decay models