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Essential Concepts
A Preview of Calculus
- Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative. Calculating a derivative requires finding a limit.
- Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.
- Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.
Introduction to the limit of a function
- A table of values or graph may be used to estimate a limit.
- If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
- If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.
- We may use limits to describe infinite behavior of a function at a point.
Key Equations
- Slope of a Secant Line
msec=f(x)−f(a)x−a - Average Velocity over Interval [a,t]
vavg=s(t)−s(a)t−a - One-Sided Limits
limx→a−f(x)=L
limx→a+f(x)=L - Intuitive Definition of the Limit
limx→af(x)=L
Glossary
- infinite limit
- A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a
- intuitive definition of the limit
- If all values of the function f(x) approach the real number L as the values of x(≠a) approach a, f(x) approaches L
- one-sided limit
- A one-sided limit of a function is a limit taken from either the left or the right
- vertical asymptote
- A function has a vertical asymptote at x=a if the limit as x approaches a from the right or left is infinite
Study Tips
The Tangent Problem and Differential Calculus
- Practice sketching secant lines for various functions and intervals.
- Visualize how secant lines approach the tangent line as points get closer.
- When working with position functions, relate concepts of average and instantaneous velocity to secant and tangent lines.
- Practice calculating average velocities over smaller and smaller intervals.
The Area Problem and Integral Calculus
- Practice approximating areas using different numbers of rectangles. Understand the relationship between smaller rectangles and better approximations.
- Visualize how the sum of rectangle areas approaches the true area as widths decrease.
- Practice setting up integrals for various shaped regions.
The Definition of a Limit
- Practice estimating limits using both tables and graphs.
- Be aware that a function can be defined at a point but have a different limit there.
- Look for oscillating behavior when suspecting a limit might not exist.
- Remember that limits describe behavior near a point, not necessarily at the point.
- Use basic limit properties to simplify complex limit problems.
One-Sided and Two-Sided Limits
- When evaluating limits, always consider both left-hand and right-hand approaches.
- For piecewise functions, use the appropriate piece for each one-sided limit.
- Create tables of values approaching the point of interest from both sides.
- Remember that a function can be continuous from one side but not the other.
- When solving limit problems, explicitly state both one-sided limits before concluding about the two-sided limit.
Infinite Limits
- Remember that infinite limits are not actual limits but descriptions of behavior.
- When finding vertical asymptotes, check both sides of the potential asymptote.
- Use factoring to identify potential vertical asymptotes in rational functions.
- Distinguish between vertical asymptotes and holes in rational functions.