1. Calculus I (MAT 145) Dr. Day Monday February 25, 2019
Derivative Shortcuts (Ch 3)
The Chain Rule!
Using and Applying Derivatives
WA Quiz #6: opens Friday, due Sunday night
Test #2: Friday, March 1: STV 308
Monday, February 25, 2019
MAT 145

2. Derivatives of Implicitly Defined Functions (3.5)
An implicitly defined function is:
A function whose relation among the variable is given by an equation for which the function has not been explicitly stated.
In the equation x2 + y2 = 25, y is an implicit function of x because the equation doesn’t explicitly express y in terms of x.
Friday, February 22, 2019
MAT 145

3. Implicit Differentiation
Find dy/dx when
Take derivative of each term. Use chain rule.
X changes identically to itself; dx/dx=1.
Solve for dy/dx.
Simplify.
Friday, February 22, 2019
MAT 145

4. Derivatives of Implicitly Defined Functions (3.5)
To calculate the derivative of an implicitly defined function:
Assume a functional connection among the variables. If x and y are present, assume y is a function of x.
Calculate the derivative of each term in the equation. Because we don’t explicitly know how y is determined by x, when calculating the derivative of y, we use the chain rule and write dy/dx as the derivative of y.
After determining term-by-term derivatives, carry out all necessary algebra steps to isolate dy/dx. That’s our goal!
Friday, February 22, 2019
MAT 145

5. Implicit Differentiation
Find dy/dx.
Friday, February 22, 2019
MAT 145

6. Derivatives of Logarithmic Functions (3.6)
What is the derivative of the natural log function y = ln(x)?
Differentiate implicitly.
Solve for dy/dx.
Substitute & simplify.
Friday, February 22, 2019
MAT 145

7. Derivatives of logarithmic functions
Friday, February 22, 2019
MAT 145

8. Derivatives of Logarithmic Functions (3.6)
Now apply this to other log functions:
Friday, February 22, 2019
MAT 145

9. Remember log rules
Friday, February 22, 2019
MAT 145

10. Derivatives of Logarithmic Functions (3.6)
And extend this to other functions:
Friday, February 22, 2019
MAT 145

11. Logarithmic differentiation
Friday, February 22, 2019
MAT 145

12. Position, velocity, acceleration
Velocity: rate of change of position
Acceleration: rate of change of velocity
Velocity and acceleration are signed numbers. Sign of velocity (pos./neg.) indicates direction of motion (right/left or up/down)
When velocity and acceleration have the same sign (both pos. or both neg.), then object is speeding up. This is because object is accelerating the same direction that the object is moving.
When velocity and acceleration have opposite sign (one positive and one negative), then object is slowing down. This is because object is accelerating the opposite direction that the object is moving.
Friday, February 22, 2019
MAT 145

13. A rabbit moves within a straight horizontal tube according to the position function
such that s(t) is measured in feet and t in seconds. When s(t) is positive, the rabbit is to the right of some arbitrary point labeled 0.
Calculate the rabbit’s velocity and acceleration functions over the interval from 0 to 6 seconds.
Determine the times from 0 to 6 seconds when the rabbit is moving to the right and when it is moving to the left. Explain your response with justification.
Determine when the rabbit is speeding up and when it is slowing down. Explain your response with justification.
Solution Guide
Friday, February 22, 2019
MAT 145

14. Friday, February 22, 2019
MAT 145

15. Friday, February 22, 2019
MAT 145

16. Implicit differentiation with respect to another variable
Suppose x, y, and z are RELATED in the following way:
And, furthermore, suppose that EACH of x, y, and z are RELATED, in some unknown way, to another VARIABLE, t:
How are each of x, y, and z changing, WITH RESPECT TO t?
Friday, February 22, 2019
MAT 145

17. Related Rates
Suppose a right triangle is growing over time. We can implicitly differentiate an equation that relates the measures of the sides to find a relationship between the rates at which the sides are changing.
dx/dt represents the rate at which the side labeled x is changing with respect to time.
dy/dt represents the rate at which the side labeled y is changing with respect to time.
dz/dt represents the rate at which the side labeled z is changing with respect to time.
Friday, February 22, 2019
MAT 145

18. Enlarging Circle
A stone is dropped in a pond and a circle ripples out from that point. The first circle’s radius increases at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 30 cm?
Vars: r, A; Given: dr/dt = 2 cm/s; Requested: dA/dt, when r=30 cm
Relationship between variables
Implicit differentiation with respect to time t leads to…
Relationship between rates:
Related Rates!
Friday, February 22, 2019
MAT 145
Ripples on the Pond

19. Enlarging Circle
A stone is dropped in a pond and a circle ripples out from that point. The first circle’s radius increases at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 30 cm?
Interpret: The area of the first circle is increasing at the rate of 120π cm2/sec, when the radius of the first circle is 30 cm.
Related-rates equation
Substitute known values for rates and variables
Solve for desired rate
Friday, February 22, 2019
MAT 145

20. Growing balloon
Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
Vars: r, V; Given: dV/dt = 100 cm3/s; Requested: dV/dt, when d = 50 cm (or r=25 cm)
Relationship between variables
Implicit differentiation with respect to time leads to…
Relationship between rates
Friday, February 22, 2019
MAT 145

21. Growing balloon
Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
Interpret: The radius of the balloon is increasing at 1/1ooπ cm/s, when the diameter of the balloon is 50 cm.
Related rates equation
Substitute known values for rates and variables (Note: d=2r)
Solve for desired rate
Friday, February 22, 2019
MAT 145

22. Ladder problem
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?
Friday, February 22, 2019
MAT 145

23. Conical water tank
A water tank has the shape of an inverted cone with a base radius 2 m and height 4 m. If water is being pumped into the tank at 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep.
Friday, February 22, 2019
MAT 145

24. Two cars heading toward intersection
Car A is traveling west at 50 mi/hr and car B is traveling north at 60 mi/hr. Both are headed for the intersection of the two roads. At what rate are the cars heading for each other when car A is 0.3 mi and car B is 0.4 mi from the intersection? Note: Read “rate cars heading for each other” as “rate at which direct line distance between them is decreasing.”
Friday, February 22, 2019
MAT 145

25. Related-Rates Strategy
Identify variables, given values and rates, and requested rate at a specific moment in time (perhaps indicated by specific value of variable).
Relate given variables.
Implicitly differentiate to find relationship among the rates.
If necessary, solve associated problem to find needed auxiliary relationships.
Substitute and solve for requested rate at specific time.
Report and interpret requested rate as increasing/decreasing and provide units!
Friday, February 22, 2019
MAT 145

26. Related Rates (3.9) Illustrations and Examples
Temple University RR site (discussion and animation)
Kelly’s RR notes (solutions and animations)
Melting Snowball discussion and animation)
More Temple University animations
Paul’s Online Math Notes: RR (worked examples)
More Relatives RR (worked examples)
Friday, February 22, 2019
MAT 145