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AP Calculus AB: 3.3.2 The Derivative of the Square Root Function

Mathematics16 CardsCreated 3 months ago

This content covers the derivative of the square root function, including how to compute it, interpret it as an instantaneous rate of change, and apply it to motion problems. It also includes examples involving limits, evaluating derivatives at specific points, and solving for velocity.

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The Derivative of the Square Root Function

• f(x) = √x is f’(x)=1/2√x
The derivative of is .
• To find the instantaneous rate of change of an object at a specific time, substitute the time into the derivative of the position function.
• To find the time when an object is moving a particular speed, set the derivative equal to that speed and solve for the independent variable.

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Key Terms

Term
Definition

The Derivative of the Square Root Function

• f(x) = √x is f’(x)=1/2√x
The derivative of is .
• To find the instantaneous rate of change of an object at a specific time, substitute the ...

note

  • To find the instantaneous rate of change given the position function, start by finding the derivative.

  • To find the derivativ...

Suppose a particle’s position is given by f(t)=√t+3,where t is measured in seconds and f(t) is given in centimeters. What is the velocity of the particle when t=1?

1/4 cm/sec

This limit represents the derivative of some function f at some number a. lim Δx→0 √2+Δx−√2 / ΔxWhat is f(x)?

f(x)=√x

Find the derivative of f(x)=√x/5.

f′(x)=1/2√5x

Suppose f(x)=√x.  What is the instantaneous rate of change in f when x=2?

1/2√2

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AP Calculus AB: 3.3.2 The Derivative of the Square Root Function
TermDefinition

The Derivative of the Square Root Function

• f(x) = √x is f’(x)=1/2√x
The derivative of is .
• To find the instantaneous rate of change of an object at a specific time, substitute the time into the derivative of the position function.
• To find the time when an object is moving a particular speed, set the derivative equal to that speed and solve for the independent variable.

note

  • To find the instantaneous rate of change given the position function, start by finding the derivative.

  • To find the derivative of the square root function, you will have to multiply the numerator and the denominator by the conjugate of the numerator.

  • Evaluate the derivative at a specific time to find the instantaneous rate of change at that time.

  • To find when an object is moving at a particular speed, set the derivative equal to that rate.

  • Solve the derivative for the independent variable.

Suppose a particle’s position is given by f(t)=√t+3,where t is measured in seconds and f(t) is given in centimeters. What is the velocity of the particle when t=1?

1/4 cm/sec

This limit represents the derivative of some function f at some number a. lim Δx→0 √2+Δx−√2 / ΔxWhat is f(x)?

f(x)=√x

Find the derivative of f(x)=√x/5.

f′(x)=1/2√5x

Suppose f(x)=√x.  What is the instantaneous rate of change in f when x=2?

1/2√2

Find the derivative of f(x)=3√x/4.

f′(x)=3/8√x

Suppose f(x)=√x^2+5. Find the equation of the line tangent to f(x)at (−2,3).

y−3=−2/3(x+2)

Find the derivative of f(x)=√x/4.

f′(x)=1/8√x

Suppose f(x)=√x+2. What is the slope of the line tangent to f when x=3?

m=1/2√5

Suppose f(x)=√2x−1. What is the slope of the line tangent to f when x=5?

m=1/3

Suppose a particle’s position is given by f(t)=√t+3, where t is measured in seconds and f(t) is given in centimeters. At what time is the velocity of the particle equal to 1/4?

t = 1

Suppose a particle’s position is given by f(t)=√2t+5,where t is measured in seconds and f(t) is given in centimeters. What is the velocity of the particle when t=2?

1/3 cm/sec

Suppose f(x)=√x−3.  What is the domain of f′(x)?

{x|x>3}

Suppose f(x)=√x+1. Find the equation of the line tangent to f(x)at (3, 2).

y−2=1/4(x−3)

Suppose a particle’s position is given by f(t)=3√t^2+5,where t is measured in seconds and f(t)is given in centimeters. At what time is the velocity of the particle equal to 2?

t = 2