Quiz 4 Fall 2006 | 2005 - 2007 | Www.ilearnmath.net
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Quiz 4 Fall 2006 | 2005 - 2007 | Www.ilearnmath.net as PDF for free.
More details
- Words: 391
- Pages: 1
AP Calculus AB :: 2006-2007 :: Shubleka
Name______________________________ Page 1
Quiz Four: Shortcuts to Differentiation 1. Prove the following: (5 points) i) The first derivative of an even function is an odd function. ii) The first derivative of an odd function is an even function. 2. The theory of relativity predicts that an object whose mass is m0 when it is at rest will appear heavier when moving at speeds near the speed of light. When the object is moving at speed v , its mass is given by m a) Find
m0
1 v 2 / c 2
, where c is the speed of light.
(10 points)
dm . dv
b) What does
dm tell you? dv
3. Find the first derivative of the given function. i)
f ( x) 3x e 4
ii)
g ( x) e( x1)
(10 points)
x
2
2
4. The rate of change of population depends on the current population, P, and is given by:
dP kP L P , for positive constants k , L. (10 points) dt a) For what nonnegative values of P is the population increasing? Decreasing? For what values of P does the population remain constant? Explain? b) Find P "(t ) as a function of P. Hint: Think of P as P(t ). 5.
(10 points) a) The curve y 11x2 is called a witch of Maria Agnesi. Find an equation of the tangent to this
curve at the point 1, 12 . Illustrate your answer by sketching the curve and the tangent on the same graph.
b) The curve y 1xx2 is called a serpentine. Find an equation of the tangent to this curve at the point 3,0.3. Illustrate your answer by sketching the curve and the tangent on the same graph. 6. Find a cubic function y ax3 bx2 cx d whose graph has horizontal tangents at the points
2,6and 2,0. (5 points)
Bonus Question Show that the curve y 6 x3 5x 3 has no tangent line with slope 4.
The Mean Value Theorem is the midwife of calculus -- not very important or glamorous by itself, but often helping to delivery other theorems that are of major significance. Purcell, E. and Varberg, D.
Name______________________________ Page 1
Quiz Four: Shortcuts to Differentiation 1. Prove the following: (5 points) i) The first derivative of an even function is an odd function. ii) The first derivative of an odd function is an even function. 2. The theory of relativity predicts that an object whose mass is m0 when it is at rest will appear heavier when moving at speeds near the speed of light. When the object is moving at speed v , its mass is given by m a) Find
m0
1 v 2 / c 2
, where c is the speed of light.
(10 points)
dm . dv
b) What does
dm tell you? dv
3. Find the first derivative of the given function. i)
f ( x) 3x e 4
ii)
g ( x) e( x1)
(10 points)
x
2
2
4. The rate of change of population depends on the current population, P, and is given by:
dP kP L P , for positive constants k , L. (10 points) dt a) For what nonnegative values of P is the population increasing? Decreasing? For what values of P does the population remain constant? Explain? b) Find P "(t ) as a function of P. Hint: Think of P as P(t ). 5.
(10 points) a) The curve y 11x2 is called a witch of Maria Agnesi. Find an equation of the tangent to this
curve at the point 1, 12 . Illustrate your answer by sketching the curve and the tangent on the same graph.
b) The curve y 1xx2 is called a serpentine. Find an equation of the tangent to this curve at the point 3,0.3. Illustrate your answer by sketching the curve and the tangent on the same graph. 6. Find a cubic function y ax3 bx2 cx d whose graph has horizontal tangents at the points
2,6and 2,0. (5 points)
Bonus Question Show that the curve y 6 x3 5x 3 has no tangent line with slope 4.
The Mean Value Theorem is the midwife of calculus -- not very important or glamorous by itself, but often helping to delivery other theorems that are of major significance. Purcell, E. and Varberg, D.
