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Calculus I

- Section 2.7 Derivatives and Rates of Change

Section 2.7 Derivatives and Rates of Change

- Def Let f be a function that is continuous on

a small neighborhood of a point P on the graph of

f, P(a, f(a)). Then to determine the slope of the

tangent line to the graph of f at P, we start by

considering a point Q(x, f(x)) close to P, with x

a. Then the slope of the secant line through

PQ

Section 2.7 Derivatives and Rates of Change

f

.

Q(x, f(x))

Then the tangent line to the curve of y f(x)

at the point P(a, f(a)) is the line through P

with slope provided the limit exists.

.

P(a, f(a))

Section 2.7 Derivatives and Rates of Change

- Example 1
- Use the above definition to find an equation

of the tangent line to the graph of - f(x) 3x2- 2x 1 at the point P(2, 9).

Section 2.7 Derivatives and Rates of Change

- Example 2
- Use the above definition to find an equation

of the tangent line to the graph of - f(x) vx at the point P(16, 4).

Section 2.7 Derivatives and Rates of Change

f

.

Q(ah, f(ah))

Then the tangent line to the curve of y f(x)

at the point P(a, f(a)) is the line through P

with slope provided the limit exists.

.

P(a, f(a))

Section 2.7 Derivatives and Rates of Change

- Example 3
- Use the above definition to find an equation

of the tangent line to the graph of - f(x) at the point P(1, 3).

Section 2.7 Derivatives and Rates of Change

- Def The derivative of a function f at a number

a, denoted by f(a) is - If this limit exists.
- So, f(a) is the slope of the tangent line at x

a, - the instanteneous rate of change of f at x a.

Section 2.7 Derivatives and Rates of Change

- Example 4
- Use the above definition to find f(a)
- f(x)

Section 2.7 Derivatives and Rates of Change

- Example 5
- If a rock is thrown into the air with a

velocity of 40 ft/s, its height (in feet) after t

seconds is given by y 40t 16t2. Find the

velocity when t 2. - RECALL Average Velocity
- Instantaneous Velocity v(a) f(a)
- Where f is the position function.

Section 2.7 Derivatives and Rates of Change

- Reference
- Stewart, Single Variable Calculus
- Early Transcendentals

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