An Introduction to Calculus

Calculus•Study of how things change

•Allows us to model real-life situations very accurately

Topics (not in order)

• Limits• Slope of a curve• Tangent Lines• Extreme values of a curve• Area under a curve/volume of a

region• Lengths• Riemann Sums (Finite & Infinite)• Rates of Change

Before Calculus After Calculus

• Find slope of a line

• Calculate average speed of a car

• Calculate area/volume of rigid geometric figures

• Find slope of a curve at a point

• Determine exact speed of a car at a particular time

• Determine area/volume of any figure under any curve/region

Two Main Problems to Solve:

I. Tangent Line Problem:How do you find the equation

of the tangent line to the graph of a curve at a particular point?

II. Area Under Curve Problem:How can you find the area underneath a curve and above the x-axis?

f(c)

f(c+x

c c+x

Secant line

Tangent line

Secant line

c+x

f(c+x

As x decreases (x→0), secant line approaches the tangent line.

sec tan0

lim ant gentxm m

tan0

lim gentx

f c x f cm

c x c

tan

0lim gentx

f c x f cm

x

a bx1 x2 x3

x x x x

a bx1 x2 x3

x x x x xxxx

x4 x5 x6 x7

As the number of rectangles increases, the sum of their areas approaches the exact area of the curve.

1

limn

ni

f a i x x

1A f a x

2A f a x x

3 2A f a x x