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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
  $m_{\sec}=\dfrac{f(x)-f(a)}{x-a}$
• Average Velocity over Interval $[a,t]$
  $v_{\text{avg}}=\dfrac{s(t)-s(a)}{t-a}$
• One-Sided Limits
  $\underset{x\to a^-}{\lim}f(x)=L$
  $\underset{x\to a^+}{\lim}f(x)=L$
• Intuitive Definition of the Limit
  $\underset{x\to a}{\lim}f(x)=L$

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 $x(\ne a)$ 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.