1. Ch. 11 – Limits and an Introduction to Calculus
11.3 – The Tangent Line Problem

2. Calculus is the study of rates of change of functions
WELCOME TO CALCULUS!!!
Calculus is the study of rates of change of functions
Recall: slope of a line determines the rate at which a line rises or falls
Today, we will use slope to study the rates of change of curves
Ex: At which point is the highest rate of change?
Observe the slopes of each line…
Draw in tangent lines to see slope
The only positive slope is at A,
so the answer is A B C A
Zoom in really, really close at 1 point. It’s basically a line, right? That’s the tangent line! D

3. As h approaches zero, we approach the true slope of the tangent line:
To find the slope of the tangent line, consider secant lines that have almost the same slope as the tangent line:
(x+h, f(x+h))
As h approaches zero, we approach the true slope of the tangent line:
(x+h, f(x+h))
(x+h, f(x+h))
(x+h, f(x+h))
(x, f(x))
(x+h, f(x+h)) x
x+h
x+h
x+h
x+h
x+h

4. Ex: Find the slope of the graph of f(x) = x2 at the point (3, 9).
Use the limit of the difference quotient from the last slide!
Ex: Find a formula for the slope of the graph of
Multiply by x(x+h)

5. The rate of change, or slope, of a function is called its derivative
The rate of change, or slope, of a function is called its derivative. It is denoted by f’(x), which is read as “f prime of x”.
Ex: Find the derivative of f(x) = 2x2 – x .

6. (Multiply by conjugate of numerator)
Ex: Find the derivative of and use it to write an equation of the tangent line through (4, 2).
Now evaluate at x=4 to find slope…
Slope = ¼
The tangent line has a slope of ¼ and passes through (4. 2), so write an equation…
I’ll use point slope form:
(Multiply by conjugate of numerator)