Derivatives
Unit 1
This Unit covers derivatives from the definition of derivative through the power rule, the product rule and the quotient rule. It also covers tangent lines and rates of change.
The videos we watch in class can be found at: http://www.calculus-help.com/tutorials
Some of the inclass examples can be found at: http://archives.math.utk.edu/visual.calculus/
All of the inclass powerpoints can be found at: http://www.hatboro-horsham.org/evans
Derivatives
Evaluate each limit.
1.
2.
3.
4.
Examples:
Use the same procedure as the previous exercises. This is merely a different notation for the same process.
Find the of the following functions.
1.
2.
3.
4.
5.
Discovering the Power Rule
1. Enter the following in your graphing calculator
2. Store the value of h as a number approaching 0, such as “.000000001”
3. Graph , look at the table of values.
4. With your group, come up with a rule for
5. Now do the same thing and try to predict the following derivatives.
a. ______
b. ______
c. ______
d. ______
e. ______
What is your conjecture for the Power Rule? ______
Practice Set #1
1. Let
a. Find [assume c is a constant]
b. Find
c. Find [assume x is a constant]
2. Differentiate the following.
Function Rewrite/Simplify Differentiate Simplify
3. Differentiate the following and simplify your final answers.
a.
b.
c.
4. Find the first and second derivatives of
5. Given Find the points of the graph where the slope is:
a. 0
b. 1
c. 4
Practice Set #2:
1. Find the equation of the line tangent to the curve at the point (3,11).
2. Find the equation of the line tangent to the curve at the point on the curve with an abscissa (x value) of 2.
3. If , find
4. Find the equation of the line that has a slope of 7 and is tangent to the curve,
5. Find the equation of the line tangent to the curve at the point on the curve located 7 units below the x-axis
6. Find the coordinates of the only second quadrant point on the curve where the line tangent to the curve will be parallel to the x-axis.
7. If , find the equation of the line tangent at the point where the curve crosses the y-axis.
8. If , find all values of x for which the slope of the curve is positive.
9. Line AB is given by . Find the equation of the line tangent to the curve and is also parallel to the line AB.
10. Find the equation of the line that has an angle of inclination of and is tangent to
c
Rate of Change (Notes)
The derivative can be used to determine the rate of change of one variable with respect to another.
Ex: Population growth, production rates, rate of water flow, velocity and acceleration.
Ex: Free fall Position function. A function, s, that gives position (relative to the origin) of an object as a function of time.
Consider: A ball dropped from a 160 foot building.
Note:
Therefore, the average velocity is
Find the average velocity over the given time intervals:
a. [1,2]
b. [1,1.5]
c. [1,1.1]
Note: Negative velocity indicates ______
Generally if, s = s(t) is the position for an object moving in a straight line, then the velocity of the object at time t is:
Find the instantaneous velocity when t = 1.1 sec.
Position Function:
Velocity Function:
Acceleration Function:
Example Set:
1. A stone is thrown vertically upward from the ground with an initial velocity of 32 ft/sec, and an equation of motion . Find
a. The average velocity of the stone during the time interval
b. The instantaneous velocity of the stone at sec and at sec
c. How many seconds will it take for the stone to reach its highest point?
d. How high will the stone go?
e. How many seconds will it take the stone to reach the ground?
f. The instantaneous velocity of the stone when it reaches the ground.
2. At t = 0, a diver jumps from a diving board that is 32 ft above the water. The position of the diver is given by the equation where s is measured in ft and t is sec.
a. When does the diver hit the water?
b. What is the diver’s velocity at impact?
Practice Set #3 - Rate of Change
1. A company finds that charging q dollars per unit produces a monthly revenue, R, Find the rate of change of R with respect to q when q = 5.
2. If the effectiveness, E, of a painkilling drug t hours after entering the blood stream is given by Find the rate of change of E with respect to t when t = 1 ; t = 3 ; t = 4
3. A diver dives from a 20-foot platform. Her initial velocity is 4 feet per second. What is her velocity when she hits the water?
4. An astronaut standing on the moon throws a rock into the air. The height of the rock is given by where s is measured in feet and t is measured in seconds. Find the acceleration of the rock and compare it with the acceleration due to gravity on earth.
5. A ball is thrown upward from ground level, and its height is given by
where s is measured in feet and t is measured in seconds.
a. Write an expression for the velocity and acceleration of the ball.
b. After how many seconds will the ball reach its maximum height and how high will it be at that time?
c. What is the velocity of the ball as it hits the ground?
6. A balloonist drops a sandbag from a balloon 160 feet above the ground.
a. Find the velocity of the sandbag after 1 second.
b. With what velocity does the sandbag hit the ground?
7. A projectile is fired directly upward from the ground with an initial velocity of 112 ft/sec.
a. What is the velocity at 3 seconds?
b. What is the maximum height that the projectile will reach?
c. What is the velocity at the instant that the projectile strikes the ground?
8. A pebble is dropped from a height of 5184 feet. Find the pebble’s velocity when it hits the ground.
9. A ball is thrown straight down from the top of a 220-foot building with an initial velocity of -22 feet per second.
a. What is it velocity after 3 seconds?
b. What is its velocity after falling 121 feet?
10. To estimate the height of a building, a stone is dropped from the top of the building. How high is the building if it strikes the ground 6.8 seconds after it is dropped?
11. A ball is dropped from a height of 100 feet. One second later another ball is dropped from a height of 75 feet.
a. Which ball hits the ground first?
b. How fast is it going when it hits the ground?
12. A man standing on top of 256 foot building throws a ball straight up in the air at a rate of 96 feet per second.
a. What is its average velocity for the first two seconds?
b. How fast is it going after 2.5 seconds?
c. What is the highest point the ball will reach?
d. How fast is the ball traveling when it hits the ground?
Practice Set #4 - Rate of Change
1. A ball is thrown upward from ground level, and its height is given by
where s is measured in feet and t is measured in seconds.
a. Write an expression for the velocity and acceleration of the ball.
b. After how many seconds will the ball reach its maximum height and how high will it be at that time?
c. What is the average rate of change in the time interval: [1, 1.4]
d. What is the velocity when t = 1 second?
e. What is the velocity the instant that the ball hits the ground?
2. If the effectiveness, E, of a painkilling drug t hours after entering the blood stream is given by Find the rate of change of E with respect to t when
a. t = 1
b. t = 2
c. t = 4
3. A balloonist drops a sandbag from a balloon 240 feet above the ground.
a. Find the velocity of the sandbag after 2 seconds.
b. With what velocity does the sandbag hit the ground?
c. What is the average rate of change in the time interval: [2, 3.5]?
4. To estimate the height of a building, a stone is dropped from the top of the building. How high is the building if it strikes the ground 7.4 seconds after it is dropped?
5. A man standing on top of 128 foot building throws a ball straight up in the air at a rate of 96 feet per second.
a. What is its average velocity for the first two seconds?
b. How fast is it going after 2.4 seconds?
c. What is the highest point the ball will reach?
d. How fast is the ball traveling when it hits the ground?
6. A ball is thrown straight down from the top of a 144 foot tall building at a speed of 48 feet per second.
a. What is the velocity of the ball at 1.5 seconds?
b. How fast is the ball going when it hits the ground?
Practice Set #5 - Rate of Change
1. A toy rocket is ejected from the top of a building 192 feet above the ground with an initial velocity of 64 feet per second.
a. What is the highest level the rocket will reach?
b. After how many seconds will it reach the ground?
c. What is the velocity of the rocket at the instant it touches the ground?
2. A ball is thrown upward with an initial velocity of 32 feet per second.
a. At what instant will the ball be at its highest point and how high will it rise?
b. What is the velocity of the ball at 2 seconds?
3. A water filled balloon is dropped from a height of 224 feet.
a. When will the balloon hit the ground and what will the velocity of the balloon be the instant it hits the ground?
b. What is the average velocity of the balloon after 2 seconds?
c. What is the instantaneous velocity by at 1.5 seconds?
4. To estimate the depth of a well a stone is dropped into the well. The stone hits the bottom of the well 5 seconds after it is dropped. How deep is the well?
5. A projectile is fired directly upward from the top of a building 160 feet high with an initial velocity of 112 feet per second.
a. What is the velocity at 2 seconds?
b. What is the maximum height that the projectile will reach?
c. If the projectile falls back onto the roof, what will be the velocity when it hits the roof?
d. If the projectile falls to the ground instead, what will the velocity be when it hits the ground?
e. How long will it take to hit the ground?
6. A spherical balloon is being blown up. The volume of a sphere is given by the formulaFind the rate of change of the volume with respect to the radius and find the rate of change of the volume of the balloon when the radius is at 3 inches.
Example Set - Higher Order Derivative Problems.
1.
2.
3.
Examples
Examples
Practice Set #6.
1. Let . Find
2. Let . Find
3. Let . Find
4. Find .
5. Let . Find
6. Find .
7. Differentiate the function .
8. Let . Find
9. Find the derivative of the function .
10. Let . Find
11. Find .
12. Write an equation for the line tangent to at the point (1,2)
13. Find any points at which the curve has a horizontal tangent line.
14. At what value(s) of x does the curve have at tangent line parallel to the line?
Examples:
Practice Set #7 - Trig Function Derivatives
1. Find
2. Find
3. Find
4. Find
5. Find
6. Find
7. Find the equation of the line tangent to the curve f(x) when
Practice Set #8 - Trig Function Derivatives
1. Find
2. Find
3. Find
4. Find
5. Find
6. Find
7. Find the equation of the line tangent to the curve f(x) when
Practice Set #9 - Higher Order Derivative Problems.
1.
2.
3.
4. Find the equation of the line tangent to the curve when f’’(x) = 6
Review
1. Find the equation of the line tangent to the curve at the point (3,3)
2. Find the equation of the line that has a slope of 5 and is tangent to the curve,
3. Find the equation of the line tangent to the curve at the point where x = 9
4. If find the value of f’(6)
5. If find the only point on the curve where the tangent line will be parallel to the x-axis.