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Introduction: Evolution and the Foundations of Biology
Overview: Inquiring About Life
An organism?s adaptations to its environment are the result of evolution
For example, a beach mouse?s light, dappled fur acts as camouflage, allowing the mouse to blend into its surroundings
Inland mice of the same species are darker in color, matching their surroundings
Evolution is the process of change that has transformed life on Earth
Figure 1.1
Figure 1.2
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 7b: Integration of Exponential Functions
Evaluate the following integrals:
6. Find the average value of the function on the interval [0, 4].
7. Write an expression and find the area of the region bounded by the curve , the x-axis, and the lines x = ln 3 and x = ln 20.
8. Water is leaking from a broken water heater at a rate in gallons per minute of .
Find the total amount of water that leaked from the water heater after 6 minutes.
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 7a: Integration by Substitution
Evaluate the following expressions:
8. Find the area under the curve on the interval .
9. Find the average value of the function on the interval .
10. A particle is moving along the x-axis so that its velocity can be found by the equation
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 6b: Vectors and Calculus
Given the following vector functions for the position of a particle, find the velocity vector:
Given the following vector functions for the position of a particle, find the acceleration vector:
A particle moves in the plane so that its position at any time is given by
Find the velocity and speed of the particle when .
Is the particle moving to the left or to the right when .
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 6a: Vectors
Given the following vectors: and :
Find the vector:
Find the vector:
Find the direction (in radians) and magnitude of the following vectors:
Given the following direction and magnitude, find a vector that represents it:
, 12
2 rad, 6
Find a unit vector in the direction of .
Find the angle between the vectors and .
Find the magnitude of the vector between the vectors and .
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 5b: Parametric Equations and Calculus
Find the derivative for the following parametric equations:
x = t2 + 1 and y = 2t3 ? t2
x = 3 sin t and y = 4cos t ? 1
Find the second derivative for the following parametric equations:
x = t2and y = t2 + 6t + 5
x = ln t and y = t2 + t
A curve C is defined by the parametric equations x = t2 + t ? 1 and y = .
Find in terms of t.
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 5a: Parametric Equations
Complete the value of table below and graph the curve for and y =
t
x
y
-2
-1
0
1
2
3
Complete the table below and graph the curve for and .
t
x
y
0
?
2?
For the following problems eliminate the parameter and find y as a function of x.
and
and
Name: ____________________ Date: ____________________
AP Calculus BC
Class Activity 4c: Differentials
Given , use a tangent line through to approximate g (1.1) to three decimal places.
Use calculus to approximate .
The measurement of an edge of a cubic box is found to be 12 inches with a possible error of inch. Use differentials to find the greatest possible error in calculating the volume of the box.
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 4b: Optimization
1. Find two numbers such that the sum of the first and three times the second is 24 and their product is a maximum.
2. Find the point on the curve that is closest to the point (4, 0)?
Hint: You will need to use the distance formula: .
3. A rectangular page is to contain 25 square inches of print. The margins on each side are 2 inches. Find the dimensions of the page such that the least amount of paper is used.
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 4a: Concavity
For the function , find and show where it is concave up on [0, 2?].
Find and show the points of inflection for the function .
Find and show the points of inflection for the function .
Given the function :
Find the critical points for p (x), i.e. the candidates for local maximums and minimums.
Use the second derivative test to show which points are maximums and which are minimums.
Given the following graph for :
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 3b: Extrema Part 2
A point moves along the line so that its position can be found by the function
Find the velocity and speed functions.
Find and verify where the particle changes direction.
At what position is the point furthest to the left?
For what interval(s) is the point moving to the left?
Given the graph below for , sketch a possible graph for :
Given the graph below for , sketch a possible graph for :
Name:________________________ Date:________________________
AP Calculus BC
Class Activity 3a: Extrema
Find the critical points for the following functions:
4. Given , find the absolute maximum and absolute minimum on the interval .
5. Given , find the absolute maximum and absolute minimum on the interval .
6. Find and show the local maximums and local minimums for the function
7. Find the interval(s) for which the following function is increasing:
8. Find and show the local minimums and local maximums for given
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