The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions, f(t), below, use F ( ) f ( t) dt to find F() and F () in terms of. 3. f(t) = t 3 4. f(t) = 6 t Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 0
Second Fundamental Theorem of Calculus Complete the table below for each function. Function F '( ) from page 77 F( ) 4t t dt Find F '( ) by applying the Second Fundamental Theorem of Calculus F( ) 4t t dt F ( ) t dt cos F ( ) t dt cos F ( ) t 3 dt F ( ) t 3 dt F ( ) 6 t dt F ( ) 6 t dt Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 585 Mark Sparks 0
Find the derivative of each of the following functions. F t dt ( ) 3 cos G ( ) e cos t dt H( ) 0 t dt Pictured to the right is the graph of g(t) and the function f() is defined to be f ( ) g( t) dt. 4. Find the value of f(0).. Find the value of f(). 3. Find the value of f '(). 4. Find the value of f '( ). 5. Find the value of f ''(). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 586 Mark Sparks 0
Given to the right is the graph of f(t) which consists of three line segments and one semicircle. Additionally, let the function g() be defined to be g( ) f ( t) dt.. Find g( 6).. Find g(6). 3. Find g '(6). 4. Find g '(). 5. Find g ''(). Give a reason for your 6. Find g ''( 4). Give a reason for your answer. Answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 587 Mark Sparks 0
Let F() = f ( t) dt, where the graph of f(t) is shown to the right. Answer the following questions. 3. Complete the following table for values of F(). 3 5 6 9 F(). On what interval(s) is f(t) positive? 3. On what interval(s) is f(t) negative? 4. On what interval(s) is F() increasing? 5. On what interval(s) is F() decreasing? Justify your answer. Justify your answer. 6. On what interval(s) is F() concave up? Justify your answer. 7. On what interval(s) is F() concave down? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 588 Mark Sparks 0
Integration of Composite Functions For each of the functions given below, find both f '( ) and f '( ) d. f() f '( ) f '( ) d f ( ) sin 3 f ( ) e cos f ( ) ln 3 f ( ) 3 Anti-differentiation by Pattern Recognition d d f ( g( )) f '( g( )) g '( ) d Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 589 Mark Sparks 0
Find each of the following indefinite integrals by pattern recognition. 3 cos3 d d 3 5 sin( ) d 3 d cos( 3 ) d 3 d e 3 3 5 d d 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 590 Mark Sparks 0
Ant-differentiation by U Substitution In each of the eight eamples above, the g '( ), or license to integrate, eisted in the integrand of f '( g( )) g '( ) d or g '( ) was attainable by multiplying by a constant. The g '( ) does not always eist and there are times when it is not attainable by multiplication of a constant. Consider the eample below. 3 ( d ) Identify the inner function, g(): What is g '( )? Is g '( ) part of the integrand? Is g '( ) attainable by multiplying the integrand by a constant? In this case, we must find the anti-derivative by a method known as U-Substitution. Here is how it works.. Let u = the inner function, g(). 4. Rewrite the entire integrand as a polynomial or polynomial type of function in terms of u. Then, anti-differentiate.. Find du and solve the equation for d. 3. Find an epression for in terms of u. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 59 Mark Sparks 0
4. Let u = the inner function, g(). 4. Rewrite the entire integrand as a polynomial or polynomial type of function in terms of u. Then, anti-differentiate. d. Find du and solve the equation for d. 3. Find an epression for in terms of u. 5 Find the value of d. Then, check the result using the graphing calculator. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 59 Mark Sparks 0
Solving Differential Equations Eamples of Variable Separable Differential Equations Given below are differential equations with given initial condition values. Find the particular solution for each differential equation.. 6 6 d and f( ) =. d 3 and f(0) = 3. d and f(0) = 4. y and f() = 3 d y Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 593 Mark Sparks 0
4 5. ( y ) and f(0) = 0 6. d y and f() = 0 d Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 594 Mark Sparks 0
The velocity of a particle is given by the function v(t) = 3t + 6 and the position at t = is 3. a. Is the particle moving to the left or right at t =? Justify your answer. b. What is the position of the particle, s(t), for any time t > 0? c. Does the particle ever change directions? Justify your answer. d. Find the total distance traveled by the particle for t = to t = 4. The acceleration of a particle moving along the ais at time t is given by a(t) = 6t. If the velocity is 5 when t = 3 and the position is 0 when t =, then the position (t) = A. 9t + B. 3t t + 4 C. t 3 t + 4t + 6 D. t 3 t + 9t 0 E. 36t 3 4t 77t + 55 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 595 Mark Sparks 0
A particle moves along the -ais so that, at any time t > 0, its acceleration is given by a(t) = 6t + 6. At time t = 0, the velocity of the particle is 9 and its position is 7. a. Find v(t), the velocity of the particle at any time t. b. For what value(s) of t > 0 is the particle moving to the right? Justify your answer. c. Find the net distance traveled by the particle over the interval [0, ]. d. Find the total distance traveled by the particle over the interval [0, ]. cos A particle moves along the ais so that its velocity is given by the function t v ( t). On the t 5 interval 0 < t < 0, how many times does the particle change directions? A. One B. Three C. Four D. Five E. Seven Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 596 Mark Sparks 0
Slope Fields Graphical Representations of Solutions to Differential Equations A slope field is a pictorial representation of all of the possible solutions to a given differential equation. Remember that a differential equation is the first derivative of a function, f '( ) or. Thus, the d solution to a differential equation is the function, f() or y. There is an infinite number of solutions to the differential equation 3. Show your work and d eplain why. For the AP Eam, you are epected to be able to do the following four things with slope fields:.. 3. 4. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 597 Mark Sparks 0
# Sketch a slope field for a given differential equation. Given the differential equation below, compute the slope for each point Indicated on the grid to the right. Then, make a small mark that approimates the slope through the point. d Given the differential equation below, compute the slope for each point Indicated on the grid to the right. Then, make a small mark that approimates the slope through the point. d y Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 598 Mark Sparks 0
# Given a slope field, sketch a solution curve through a given point. To the right is pictured the slope field that you developed for the differential equation on the previous page. d Sketch the solution curve through the point (, -). To do this, you find the point and then follow the slopes as you connect the lines. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 599 Mark Sparks 0
#3 Match a slope field to a differential equation. Since the slope field represents all of the particular solutions to a differential equation, and the solution represents the ANTIDERIVATIVE of a differential equation, then the slope field should take the shape of the antiderivative of /d. Match the slope fields to the differential equations on the net page. A. B. C. D. E. F. G. H. I. J. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 600 Mark Sparks 0
Separate the variables and find the general solution to each differential equation below to determine what the slope field should look like for each. Then, match to the graphs of slope fields on the previous page.. sin. 4 d d 3. e 4. d d 3 5. 3 6. cos d d 7. 4 8. d d 9. 0. d d Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 60 Mark Sparks 0
#4 Match a slope field to a solution to a differential equation. When given a slope field and a solution to a differential equation, then the slope field should look like the solution, or y. Match the slope fields below to the solutions on the net page. A. B. C. D. E. F. G. H. I. J. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 60 Mark Sparks 0
. y. y 3. y e 4. y 5. 3 y 6. y sin 7. y cos 8. y 9. y 0. y tan Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 603 Mark Sparks 0
Shown below is a slope field for which of the following differential equations? Eplain your reasoning for each of the choices below. Consider the differential equation to answer the following questions. d y a. On the aes below, sketch a slope field for the equation. b. Sketch a solution curve that passes through the point (0, ) on your slope field. c. Find the particular solution y = f() to the differential equation with the initial condition f(0) =. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 604 Mark Sparks 0