Calculus I

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Calculus I

• Section 2.7 Derivatives and Rates of Change

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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

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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))
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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).

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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).

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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))
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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).

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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.

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Section 2.7 Derivatives and Rates of Change

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

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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.

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Section 2.7 Derivatives and Rates of Change

• Reference
• Stewart, Single Variable Calculus
• Early Transcendentals