1.1 Preview of Calculus Objectives: -Students will understand what calculus is and how it compares with precalculus -Students will understand that the tangent line problem is basic to calculus -Students will understand that the area problem is also basic to calculus

Two Problems Led to Calculus: Problem 1) How do you find the slope of a function at one point? a) Why do we care? Instantaneous rate of change. miles time

b) How do we do it? -start with an “algebra slope” (slope between 2 points) -finish with a “calculus slope” (slope at 1 point) -as the change in x gets smaller, the slope of the secant line approaches the slope of the tangent line

Two Problems Led to Calculus: Problem 2) How do you find the area under a curve of a function? a) Why do we care? Accumulation of change.

b) How do we do it? -start with rectangles, then increase the number of rectangles. n = # of rectangles Area =

So, in both problems limits are very important! Ch 1: Limits Ch 2 and Ch 3: Problem 1 Ch 4 and Ch 7: Problem 2 Ch 5 and Ch 6: Connecting Problems 1 and 2

Ex 1 (problem 7): Consider the function and the point P(1,3) on the graph of f. a) Graph f and the secant lines passing through P(1,3) and Q(x,f(x)) for x-values of 2, 1.5, and 0.5.

Ex 1 (problem 7): Consider the function and the point P(1,3) on the graph of f. b) Find the slope of each secant line. point ( 1, 3 ) point ( 2, )m= point ( 1.5, )m= point ( 0.5, )m=

Ex 1 (problem 7): Consider the function and the point P(1,3) on the graph of f. c) Use the results of part (b) to estimate the slope of the tangent line of f at P(1,3). Describe how to improve your approximation of the slope.

Ex 2 (problem 10): a) Use the rectangles in each graph to approximate the area of the region bounded by y = sin x, y = 0, and x = π b) Describe how you could continue this process to obtain a more accurate approximation of the area.

Other Foundational Ideas: -Interpreting Graphs: d t t v

Other Foundational Ideas: -Proportional Equations: Directly Proportional Ex) Grade, g, is directly proportional to number of hours, h, studied. Inversely Proportional Ex) Distance driven, d, is inversely proportional to amount of gas in tank, g.

Other Foundational Ideas: -Functional Notation: a) f(x) for x = 2 b) f(3) – 1 c) f(3 – 1) d) 2g(x) for x = 1 e) g(2x) for x = 1 f) solve g(x) = 17 x01234 f(x) g(x)

-Graphing Functional Notation: a) f(c) = b) g(f(a)) = c) g -1 (b) = d) coord. of intersection: x cab c b a f(x) g(x)

1.2 Finding Limits Graphically and Numerically Objectives: -Students will estimate a limit using a numerical or graphical approach -Students will learn different ways that a limit can fail to exist -Students will study and use a formal definition of a limit

Limits Think: “As x gets close to limit value, what does y get close to?”

There are 3 ways to find the limit of I. Graphically – draw a graph by hand or with a calc.

There are 3 ways to find the limit of II. Numerically – table of values x f(x)

There are 3 ways to find the limit of III. Analytically – use algebra -use direct substitution

Ex 1) Using any of the 3 methods discussed, solve -try direct substitution first: -graphically: -could also do a table: ** the existence/non-existence of f(x) at x=c has no bearing on the existence of the limit.

Sometimes a limit does not exist (DNE)… 1) f(x) approaches a different y-value from the right side of c than it approaches from the left.

Sometimes a limit does not exist (DNE)… 2) f(x) increases or decreases without bound as x approaches c. Because f(x) isn’t approaching a real number, the limit does not exist.

Sometimes a limit does not exist (DNE)… 3) f(x) oscillates between two fixed values as x approaches c.

Ex 1) Practice graphing the following piecewise function:

Ex 2) Find for Ex 3) Find