3.3.2 The Derivative of the Square Root Function

Question 1

The Derivative of the Square Root Function

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 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. What is the velocity of the particle when t=1?

Answer 3

1/4 cm/sec

Question 4

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

Answer 4

f(x)=√x

Question 5

Find the derivative of f(x)=√x/5.

Answer 5

f′(x)=1/2√5x

Question 6

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

Answer 6

1/2√2

Question 7

Find the derivative of f(x)=3√x/4.

Answer 7

f′(x)=3/8√x

Question 8

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

Answer 8

y−3=−2/3(x+2)

Question 9

Find the derivative of f(x)=√x/4.

Answer 9

f′(x)=1/8√x

Question 10

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

Answer 10

m=1/2√5

Question 11

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

Answer 11

m=1/3

Question 12

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?

Answer 12

t = 1

Question 13

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?

Answer 13

1/3 cm/sec

Question 14

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

Answer 14

{x|x>3}

Question 15

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

Answer 15

y−2=1/4(x−3)

Question 16

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?

Answer 16

t = 2