Calculus involves finding properties and features of graphs, such as intercepts, maxima, minima, and slopes. Students first learn to identify these visually and then through algebraic equations alone. The AP Calculus exam will include both normal and intentionally difficult graphs and formulas to assess these concepts. Students should not be intimidated by unfamiliar notation or vocabulary, as the underlying ideas being tested will generally be straightforward.

1. Overview of Calculus

2. • Calculus is all about finding different
features/properties of graphs
• In this class, you will first learn how to identify
the features/properties by looking at the graph
• Then, you will learn how to find them using
algebra/equations when only given just a
formula (no picture).

3. Example Graph:

4. • Example of feature/property of functions that you
can find using Algebra/Equations that you
ALREADY KNOW:

5. Intercepts
𝑥 intercept𝑥 intercept𝑥 intercept
𝑦 intercept

6. Given some formula: 𝑦 = "stuff with 𝑥"
𝑥 intercepts → set 𝑦 = 0 and solve equation for 𝑥
𝑦 intercept → set 𝑥 = 0 and solve equation for 𝑦
“Alegbra/Equation way” to find intercepts:

7. • There are many other properties/features of a
function that you will find using just a formula
and Algebra/Equations (NO picture/graph at all)
in Calculus:

8. Minimum
Maximum
Maximums and Minimums
(top or bottom points)

9. Inflection Points (middle of a squiggle)
Inflection Points

10. Intervals of Increasing (going UP) and
Decreasing (going DOWN)

11. End-behavior (what the ends of the graph on
the left or right are doing)

12. Slope of Secant Lines
(line between two points on graph)

13. Slope of Tangent Lines
(line tangent to graph: touch at one point)

14. Area Under a Curve

15. • For the most part on the AP, you will just be asked to find these features
when given a formula using Algebra/Equations.
– Example: Finding a Minimum
• However, there will also be word problems in AP Calculus where what you
are asked to find is represented by one of these features on the previous
slides.
– Example: Finding a Minimum in a Word Problem:

16. Normal-Looking Graphs and Crazy-Looking Graphs
• Because AP Calculus is made by the same people who make the SAT, the whole
purpose of the test is to MESS WITH YOU.
• The test will have some normal-looking graphs & formulas, as well as some crazy-
looking graphs & formulas just for the purpose of being unnecessarily
hard/messing with you:
“Normal Graph” “Crazy Graph”

17. Other Ways the AP test will mess with you:
• Scary-looking symbols/notation:
• Asking the SAME question in many different ways:
1. “Find the derivative at 𝑥 = 2”
2. “Find the instantaneous rate of change at 𝑥 = 2”
3. “Find 𝑦′(2)”
4. “Find 𝑓′(2)”
5. “Find
𝑑𝑦
𝑑𝑥
at 𝑥 = 2”
(these 5 questions all mean the same exact thing)
• DON’T be scared by all the weird symbols and vocabulary! Trust
that every time you see something confusing in AP Calculus, it is
always there to scare you off but it ACTUALLY EASY 
lim
𝑥→∞
𝑓(𝑥) 𝑑2 𝑦
𝑑𝑥2
𝑖=1
𝑛
𝑓(𝑥𝑖)
0
1
𝑓(𝑥) etc.