Calculus Higher Derivatives

​Higher order derivatives and Motion

By John Lawhon

Multiple Choice

At which point(s) will the slopes of the tangent line be zero?

• at C only

• at points A, C and E only

• at point B and D only

• at points A and E only

Multiple Choice

Where on this graph would you find a HORIZONTAL tangent line?

• x = -3

• x = 0

• x = 3

• There are no horizontal tangent lines on this graph

Multiple Choice

The percentage grade a student receives on a test, is modeled by G(t) where t is the number of hours spent studying for the test. Interpret G'(1) = 3

• At 1 hour of studying, the grade will improve at a rate of 3% per hour.

• At 3 hours of studying, the grade will improve at a rate of 1% per hour.

• The student gains 3 percentage point per hour of study.

• The student loses 3 percentage point per hour of study.

Multiple Choice

The number of gallons of water in a storage tank at time t, in minutes, is modeled by w(t).

Interpret w'(10) = -8

• The tank loses 8 gallons per minute at 10 minutes.

• The tank gains 8 gallons per minute at 10 minutes.

• The rate of the tanks water loss is decreasing by 8 gallons per minute per minute at 10 minutes.

• The tank gains 10 gallons per minute at 8 minutes.

Multiple Choice

What is the derivative of f(x) = 2x³ - 4x + 5?

• f'(x) = 6x² - 4x

• f'(x) = 2/3x² - 4x

• f'(x) = 3x² - 4

• f'(x) = 6x² - 4

Higher Order Derivatives

When we just keep taking DERIVATIVES!

What are higher order derivatives?

• When you continue to take the derivative of a function!

• Each iteration uses the power rule

• You can take derivatives infinitely!

Multiple Choice

How many derivatives can you take of a function?

• only 3
• only 2
• only 1
• until you can no longer derive it

Multiple Choice

The first derivative of $$y=f\left(x\right)$$ is represented as

• f'(x)

• y'

• dy/dx

• none of these

Multiple Choice

The 2nd derivative of $$y=f\left(x\right)$$ also known as the derivative of the 1st derivative is represented as

• f''(x)

• y''

• $$\frac{d^2y}{dx^2}$$

• none of these

Poll

Find the second derivative of the following functions with respect to x.
y=3x²+5x-1

• 6x+5
• 3x+5
• 5
• 6

Poll

Find the second derivative of f(x) = x² + e^x  - cosx

• f"(x) = 2 + e^x + cosx
• f"(x) = 2x + e^x + cosx
• f"(x) = 2x + xe^x - cosx
• f"(x) = 2x + e^x + sinx

Multiple Choice

Find y'' for y = 3x² + 5x + cos x

• f''(x) = 6x + cos x

• f''(x) = 3x + 5 - sin x

• f''(x) = 6x + 5 + sin x

• f''(x) = 6 - cos x

Multiple Choice

If the position of a particle is represented by x(t) = -t² + 1, what is its instantaneous velocity at t = 1?  

• v = 0
• v = 1
• v = -1
• v = -2

Multiple Select

Acceleration is

• integral of velocity

• derivative of position

• derivative of velocity

• second derivative of poisiton

Multiple Select

Which of the following indicated that t=0?

• to start

• begins

• finishes

• initially

Multiple Choice

On the right means

• position is positive

• velocity is positive

• position is negative

• velocity is negative

Multiple Choice

On the left means

• position is positive

• velocity is positive

• position is negative

• velocity is negative

Multiple Choice

Above the ground means

• position is negative

• velocity is positive

• position is positive

• velocity is negative

Multiple Choice

Below the ground means

• position is negative

• velocity is positive

• position is positive

• velocity is negative

Multiple Choice

Moving to the right means

• acceleration is positive

• velocity is positive

• position is positive

• velocity is negative

Multiple Choice

Moving to the left means

• acceleration is negative

• velocity is positive

• position is negative

• velocity is negative

Multiple Choice

Moving up means

• acceleration is positive

• position is positive

• velocity is positive

• velocity is negative

Multiple Choice

Moving down means

• velocity is negative

• velocity is positive

• position is negative

• acceleration is negative

Multiple Choice

Dropped or at rest means

• Acceleration is equal to 0

• Position is equal to 0

• Velocity is equal to zero

• Acceleration is positive

Multiple Choice

Farthest to the right means

• Find the maximum of position

• Find the maximum of velocity

• Find the maximum of acceleration

• Set velocity equal to 0

Multiple Choice

Farthest to the left means

• Find the minimum of acceleration

• Find the minimum of position

• Find the minimum of velocity

• Set velocity equal to 0

Multiple Choice

Higest means

• Find the maximum of acceleration

• Find the maximum of velocity

• Find the maximum of position

• Set velocity equal to 0

Multiple Choice

Lowest means

• Find the minimum of position

• Find the minimum of acceleration

• Find the minimum of velocity

• Set velocity equal to 0

Multiple Choice

An object moves towards the origin when

• position and velocity are the same sign

• position and velocity are opposite signs

• velocity and acceleration are the same sign

• velocity and acceleration are opposite signs

Multiple Choice

An object moves away from the origin when

• position and velocity are the same sign

• position and velocity are opposite signs

• velocity and acceleration are the same sign

• velocity and acceleration are opposite signs

Multiple Choice

Speed is increasing when

• position and velocity are the same sign

• position and velocity are opposite signs

• velocity and acceleration are the same sign

• velocity and acceleration are opposite signs

Multiple Choice

Speed is decreasing when

• position and velocity are the same sign

• position and velocity are opposite signs

• velocity and acceleration are the same sign

• velocity and acceleration are opposite signs

Poll

Given the graph of f'(x), what's the interval of the graph of f(x) is increasing, select all the answers

• (-4,-3)

• (-1,1)

• (2,infinity)

• (-2,0)

Multiple Choice

If a function's FIRST derivative is negative at a certain point, what does that tell you?

• The function is increasing at that point
• The function is decreasing at that point
• The concavity of the function is up at that point
• The concavity of the function is down at that point

Multiple Choice

Over what interval(s) is f(x) decreasing?

• (-3, 1)
• (-∞, -5) ∪ (0, 2)
• (-∞, -3) ∪ (1, ∞)
• (-5, 0) ∪ (2, ∞)

Multiple Choice

If  $$f'\left(x\right)=0$$  and  $$f'\left(a\right)=0$$  changes from negative to positive at $$f'\left(x\right)$$  , then  $$x=a$$  has

• a relative maximum at x=a

• a relative minimum at x=a

• no relative extrema at x=a

• a vertical tangent line at x=a

Multiple Choice

If  $$f'\left(x\right)$$  and  $$f'\left(x\right)$$  changes from positive to negative at  $$x=a$$  , then  $$f\left(x\right)$$  has 

• A relative maximum at x=a

• A relative minimum at x=a

• No relative extrema at x=a

• A vertical tangent line at x=a

Multiple Choice

The function  $$f\left(x\right)=-x^3+3x^2-5$$  has a relative maximum value of _____________ at x=_____________.

• 0; -5

• -5; 0

• 2; -1

• -1; 2

Multiple Choice

If  $$\ f'\left(a\right)=0$$  , then you are guaranteed a relative extreme value at x=a. 

• True

• False

Multiple Choice

Which of the following could be the graph of f ' , the derivative of f ?

Match

Match the following notation:

• position
• acceleration
• velocity

• speed
• displacement

• $$s\left(t\right)$$

• $$v'\left(t\right)$$

• $$s'\left(t\right)$$

• $$\left|v\left(t\right)\right|$$

• $$s\left(b\right)-s\left(a\right)$$

Multiple Choice

A bug begins to crawl up a vertical wire at time $$t=0$$ .  The velocity $$v$$ of the bug at time $$t$$ , $$0<t<8$$ , is given by the function whose graph is shown behind this text.

At what value of $$t$$ does the bug change direction? 

• 2
• 4
• 6.5
• 7

Dropdown

A particle moves along the x-axis so that at time $$t\ge0$$ , its velocity is given by $$v(t)=−2t+8$$ .

The particle is ​

Select answer

at $$t=7$$ ​ .

Multiple Choice

Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of $$\sqrt[]{f\left(x\right)+g\left(x\right)}$$   at x = 3.

• $$\frac{1}{2\sqrt[]{10}}$$  

• $$\frac{9}{2\sqrt[]{10}}$$  

• $$\frac{9}{\sqrt[]{10}}$$  

• $$-\frac{1}{2\sqrt[]{10}}$$  

Multiple Choice

The position of a particle moving along a coordinate line is $$s=\sqrt[]{2+2t}$$   with s in meters in t in seconds. Find the particle's acceleration at t = 1 second.

• -1/16 m/sec^2

• 1/2 m/sec^2

• 1/8 m/sec^2

• -1/8 m/sec^2

Multiple Choice

Find y'' if y = 6x sinx

• $$y''=-12\cos x+6x\sin x$$  

• $$y''=6\cos x-12x\sin x$$  

• $$y''=12\cos x-6x\sin x$$  

• $$y''=-6x\sin x$$  

Multiple Choice

The number of gallons of water in a swimming pool t minutes after the pool has started to drain is $$Q\left(t\right)=50\left(20-x\right)^2$$  . How fast is the water running out after 15 minutes?

• 1250 gal/min

• 625 gal/min

• 500 gal/min

• 250 gal/min

Multiple Choice

Find the instantaneous velocity when x = 2.

$$f\left(x\right)=x^3\ \sqrt[]{x^2-1}$$  

• $$\frac{4}{\sqrt[]{3}}+12\sqrt[]{3}$$  

• $$28\sqrt[]{3}$$  

• $$4\sqrt[]{3}$$  

• $$\frac{16}{\sqrt[]{3}}+12\sqrt[]{3}$$