Lesson 7: Applications of the Derivative

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Presentation transcript:

Lesson 7: Applications of the Derivative MATH 1314 Lesson 7: Applications of the Derivative

Meaning of a Derivative:

f'(x) f'(4)

tangent(3,f(x))

Root(f'(x))

intersect(f'(x),g(x))

f''(x) f’’ (5)

Popper 4 Find the slope of the tangent line at the given x-value. f(x) = 3x2 – x at x = -1 -5 B. -1 C. -7 D. 6 2. f(x) = -x2 – 2x at x = 0 -2 B. -1 C. 2 D. 1 3. f(x) = x3 + x2 at x = 2 A. -2 B. 10 C. 2 D. 16

Popper 4 Continued 4. Find all values of x for which the slope of the tangent line to the function is horizontal: f(x) = -x2 – 2x -2 B. -1 C. 2 D. 1 5. Find all values of x for which the slope of the tangent line to the function is horizontal: f(x) = x2 + 4x -4 B. -2 C. 0 D. 4 6. Find all values of x for which the slope of the tangent line to the function is horizontal: f(x) = -10x5 A. -10 B. -1 C. -50 D. 0

Word Problems:

Instantaneous vs. Average Rate of Change

Position, Velocity, and Acceleration

Free Fall

root(f(t)) f'(6.352)

f'(3) -12 Falling

2.625 root(f'(t))

-32 f''(t)

Popper 5: The function d(t) = 4t2 + 32t models the distance covered by a car t seconds after starting from rest. Find the distance covered by the car after 3 seconds. Find the distance covered by the car after 5 seconds. 150 B. 180 C. 108 D. 260 E. 132 3. Find the velocity of the car when t = 4. Find the velocity of the car when t = 7. A. 88 B. 64 C. 48 D. 60 E. 55

Popper 5 Continued: The function d(t) = -t3 + 15t2 + 32t models the distance covered by a car t seconds after starting from rest. 5. Find the acceleration of the car when t = 3. 6. Find the acceleration of the car when t = 4. A. 63 B. 6 C. 75 D. 12 E. 204