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How do you evaluate z x sin(X2) DX?
EXAMPLE 8.1.3 Evaluate Z x sin(x2) dx. First we compute the antiderivative, then evaluate the definite integral. Let u = x2 so du = 2x dx or x dx = du/2. Then cos(4). 2 2 A somewhat neater alternative to this method is to change the original limits to match the variable u. Since u = x2, when x = 2, u = 4, and when x = 4, u = 16. So we can do this:

How many ft/sec is a t 0 T18 seconds?
1 : limits and 55 ft/sec at time t = 0. For 0 t18 seconds, the cars acceleration a()t , in ft/sec2, is the piecewise linear function defined by the graph above.

What is Z sec2 u u du?
sec2 u sec2 u du = Z sec3 u du. In problems of this type, two integrals come up frequently: Z sec3 u du and Z sec u du. Both have relatively nice expressions but they are a bit tricky to discover. First we do du. tan u + sec2 u du, exactly the numerator of the function we are integrating. Thus

What if a student incorrectly solves a dy dx equation?
The student correctly solves the equation to find the time at which the line tangent to the path of the particle is horizontal and earned the second point. The student incorrectly reasons that the motion of the particle at the t-value presented can be determined from dy dx and did not earn the third point.

[PDF File]Techniques of Integration - Whitman College
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sec2 xdx = tanx+ C Z secxtanxdx = secx+C Z 1 1+ x2 dx = arctanx+ C Z 1 √ 1− x2 dx = arcsinx+ C 8.1 Substitution Needless to say, most problems we encounter will not be so simple. Here’s a slightly more complicated example: ﬁnd Z 2xcos(x2)dx. This is not a “simple” derivative, but a little thought reveals that it must have come from

[PDF File]AP CALCULUS BC 2010 SCORING GUIDELINES - College Board
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Find each of the following. (i) The two values of t when that occurs (ii) The slopes of the lines tangent to the particle’s path at that point The y-coordinate of that point, given y 2 1 ( ) = 3 e Speed = 2 2 ( ′ 3 y ( ) ) + ( ′ 3 2.828 meters per second ( ) ) = x ′ t 2 t 4 ( ) = − 4 Distance = ∫ 0 2 2 t 4 2 te t − 3 ( − ) + − (

[PDF File]2001 AP Calculus BC Scoring Guidelines - College Board
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and 3sin 2 dy t dt for 03 t. At time t = 2, the object is at position (4,5). (a) Write an equation for the line tangent to the curve at (4,5). (b) Find the speed of the object at time t = 2. (c) Find the total distance traveled by the object over the time interval 01 t. (d) Find the position of the object at time t = 3. (a) 2 3 3sin cos dy t dx ...

[PDF File]AP 2006 SCORING GUIDELINES (Form B) Question 2 - College Board
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sec 1 tan sin t tt dy dy dt e dx dx ee dt − −− == = ()2, 3 ()1 1 2.780 sin dy dx e− − == or 2.781 ()1 1 32 sin yx e− += − 2 : ()2, 3 1 : 1 : equation of tangent line dy dx − ⎧ ⎪ ⎨ ⎪ ⎩ (b) x′′ ()1 0.42253,=− y′′ ()1 0.15196=− a()10.423,0.152=− − or −−0.422, 0.151 . speed ()()()() =+ =sec tan 1.138ee ...

[PDF File]AP CALCULUS BC 2012 SCORING GUIDELINES - College Board
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AP CALCULUS BC 2012 SCORING GUIDELINES Question 2 For t 0, a particle is moving along a curve so that its position at time t is x t y t t ( ( ) , . At time ( ) ) = 2, the dx particle is at position 1, 5 . It is known that ( ) dt = dy 2 and e dt sin 2 t . = Is the horizontal movement of the particle to the left or to the right at time t = 2 ?

[PDF File]The Fundamental Theorem of Calculus. - Saint Louis University
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The Fundamental Theorem of Calculus. The two main concepts of calculus are integration and di erentiation. The Fundamental Theorem of Calculus (FTC) says that these two concepts are es- sentially inverse to one another. The fundamental theorem states that if F has a continuous derivative on an interval [a; b], then b F0(t)dt = F(b) F(a):

Dx t 1 t 2 sec