AP Calculus AB

Assignment Guide

Chapter 2 (Part B)

Date / Section/Skills / Assignment
10/7
10/8 / 2.5
  • Compute derivatives by implicit differentiation
/ Day 1:
P. 146 1 – 27 o
Day 2:
p. 146 33, 37, 41 (a only), 45, 47, 49, 53, 57, 74
10/9
10/10
10/11
10/15 / 2.6
  • Solve related rate problems
  • Geometric related rates
  • Pythagorean related rates
  • Angular related rates
/ Day 1:
p. 154 & 155 13, 16-22
Day 2:
p. 155 & 156 25, 26, 28-32, 42-46
Day 3:
No HW
**Quiz – Implicit Differentiation**
Day 4:
Take Home Quiz – Related Rates
10/16
10/17 /
  • Review – implicit differentiation & related rates
/ Day 1:
p. 160 103, 107, 109, 112-115
p. AP 2-1 8-10
Day 2:
HW – WS – Related Rates Review
10/18 / **Test 2.5/2.6**

Learning Target #8:I can use implicit differentiation to find the derivative of a curve.

I can distinguish when to utilize explicit versus implicit differentiation.

I can find the slope or equation of a tangent line of implicitly defined curves.

I can determine where an implicitly defined curve has horizontal/vertical tangents.

I can find higher order derivatives of implicitly defined curves.

Learning Target #9:I can use related rates to solve real-life problems.

I can identify the appropriate related dimensions/variables and the primary equation which relates them, including well-known geometric formulas, right triangle formulas, and trigonometric ratio equations

.

I can correctly differentiate the primary equation with respect to time.

AP Calculus AB

Notes –Section 2.5 ( Implicit Differentiation)

Day 1

Learning Target: Students will be able to use implicit differentiation to find the derivative of implicitly defined functions.

Explicit form vsImplicit form

Steps for implicit differentiation:

  1. Differentiate both sides of the equation with respect to x (don’t forget to add or when taking the derivate of any terms with a y.
  2. Collect all terms involving or on the left side of the equation and move all other terms to the right side of the equation.
  3. Factor or out of the left side of the equation.
  4. Solve for or by dividing both sides of the equation by the left-hand factor that does not contain or .

Example 1.Example 2.

Find given that Find given that

Example 3.

Find given that

Example 4.Example 5.

Find given that Find given that .

Example 6.

Find the equation of the tangent line to at (1, 1).

AP Calculus AB

Notes –Section 2.5 (Implicit Differentiation)

Day 2

Learning Target: Students will be able to use implicit differentiation to find the second derivative of implicitly defined functions.

Find for each of the following problems.

  1. 2. at

To find ,

  1. Find
  2. Calculate .
  3. Substitute into as needed. Simplify.

Example 1.

Find if

Example 2.

Find if

Example 3.

Find if

Example 4.

Given the following graph of a function , graph its derivative. Create a sign chart to justify.


AP Calculus AB

Notes – Related Rates

Learning Target: Students will be able to solve related rates problems.

Find the derivative of the following with respect to time.

1.2.

3.4.

Solving Related Rates Problems:

• Find an equation that ties your variables together.

•You may now plug in any constant value. Do not plug in any value that changes.

• Differentiate your equation with respect to time.

• Plug in all variables. If you have more than one unknown, you will most likely use your original equation to find the missing value.

• Label your answers in terms of the correct units and be sure that you answered the question asked!

(TI-Nspire/Navigator )

Problem 1 – Example & Explanation
Water is draining from a cylindrical tank at 4 liters/seconds. If the radius of the tank is 2centimeters, how fast is the surface dropping?
Step 1:Assign variables, list given information, and determine the unknown(s).
  • Variables:
  • Given information:
  • Unknown(s):
Step 2:Write a formula relating given(s) and unknown(s) for a cylindrical tank.
Step 3:Differentiate both sides of the equation from Step 2 with respect to t to find the related rates.
  • Is this problem solved using the Product Rule? Explain.
  • Use implicit differentiation to differentiate the equation. Show your work.
Step 4:Evaluate—substitute and answer the question being asked!
  • How fast is the surface dropping when the radius is 2 cm?

Problem 2 – Additional Example & Explanation
Two cars leave at the same time, one traveling east at 15 units/hour and the other traveling north at 8 units/hour. At what rate is the distance between them increasing when the car going east is 30 units from the starting point?
Step 1:Assign variables, list given information, and determine the goal.
Step 2:What equation relates what is known and what you want to find?
Step 3:Implicitly differentiate both sides of the equation from Step 2 with respect to t to find the related rates.
Step 4:Substitute to evaluate.
  • Show these key steps
  • What is the correct answer to the original problem?

Problem 3 – Homework/Extension
1.A spherical bubble is being blown up. The volume is increasing at the rate of 9 mm3 per second. At what rate is the radius increasing when the radius is 3 mm?
2.A point moves along the curve y= –0.5x2+ 8 in such a way that the y value is decreasing at the rate of 2 units per second. At what rate is x changing when x= 4?
3.A particle moves on the curve such that = 6. Find the instantaneous rate of change of xwith respect to t when x = 2.
4.A balloon is submerged in liquid nitrogen. The balloon’s diameter contracts when it is cooled. The diameter of the sphere is decreasing at a rate of 4 cm/s, how fast is the surface area changing when the radius is 10 centimeters?
5.Two trains leave the station at the same time with one train traveling south at 20 mph and the other traveling west at 33 mph. How fast is the distance between the trains changing after 3 hours?
6A cylindrical tumbler with a radius of 3 cm has its height increasing at a rate of 2.5 cm/sec. Find the rate of change of the volume of the cylinder when the height is 12.56 cm.

Geometric Related Rates

Example 1

A right circular cylinder with a height of 10 feet and radius of 8 feet has dimensions that are changing. Write formulas for the volume and surface area of the cylinder and the rate at which they change.

  • The radius is growing at 2 ft/min and the height is shrinking at 3 ft/min
  • Change of volume:
  • Change of surface area:
  • The radius is decreasing at 4 ft/min and the height is increasing at 2 ft/min
  • Change of volume:
  • Change of surface area:

Example 2:

An oil tank spills oil that spreads in a circular pattern whose radius increases at the rate of 50 ft/min. How fast are both the circumference and area of the spill increasing when the radius of the spill is a) 20 ft and b) 50 ft?

Pythagorean Related Rates

Example 3:

A 15 foot ladder is resting against the wall. The bottom is initially 10 ft away from the wall and is being pushed towards the wall at a rate of 1/4 ft/sec. How fast is the top of the ladder moving up the wall when the base of the ladder is 7 ft from the wall?

x’(t) = dx/dt = -1/4

y’(t) = dy/dt = ?

Example 4;

Two people start out 50 feet apart. One of them starts walking north at a rate so that the angle shown in the diagram is changing at a constant rate of 0.01 rad/min. At what rate is distance between the two people changing when = 0.5 radians?

Angular Related Rates

Example 5:

Tracking a rocket: A spy tracks a rocket through a telescope to determine its velocity. The rocket is traveling vertically from a launching pad located 10 km away, as in the figure. At a certain moment, the spy’s instruments show that the angle between the telescope and the ground is equal to and is changing at a rate of 0.5 rad/min. What is the rocket’s velocity at that moment?

Example 6:

Revolving Light. Officers from the Crystal Lake police department apprehended a group of students who were playing a game of fugitive. The revolving light on the police car located 10 m from the garage turns with a constant angular velocity. With what velocity does the light revolve if the light moves along the garage door at the rate of 15 meters per minute when the beam makes an angle of with the garage door?

Related Rates

Extra Problems

1.A searchlight rotates at a rate of 3 revolutions per minute. The beam hits a wall located 10 miles away and produces a dot of light that moves horizontally along the wall. How fast is this dot moving when the angle between the beam and the line through the searchlight perpendicular to the wall is ?

2.A conical tank has height 2 m and radius 2 m at the top. Water flows in at a rate of 2 m3/min. How fast is the water level rising when it is 2 m deep?

3.A road perpendicular to a highway leads to a farmhouse located 1 mile away. An automobile travels past the farmhouse at a speed of 60 mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 3 miles past the intersection of the highway and the road?

AP Calculus ABName

Related RatesDate

Practice

1.A spherical hot air balloon is being inflated. If air is blown into the balloon at the rate of 2ft3/sec,

a. Find how fast the radius of the balloon is changing when the radius is 3 ft.

b. Find how fast the surface area is increasing at the same time.

2.A 12 foot ladder stands against a vertical wall. If the lower end of the ladder is being pulled away from the wall at the rate of 2 ft/sec,

a. How fast is the top of the ladder coming down the wall at the instant it is 6 ft above the ground?

b. How fast is the angle of the elevation of the ladder changing at the same instant?

3.A boy flies a kite that is 120 ft directly above his hand. If the wind carries the kite horizontally at the rate of 30 ft/min, at what rate is the string being pulled out when the length of the string is 150 ft?

4.A revolving light located 5 miles from a straight shore line turns with a constant angular velocity. What velocity does the light revolve if the light moves along the shore at the rate of 15 miles per minute when the beam makes an angle of 60° with the shore line?

5.How fast does the water level drop when a cylindrical tank of radius 6 feet is drained at the rate of 3 ft3/min?