Drill problems on derivatives and antiderivatives 1 Derivatives Find the derivative of each of the following functions (wherever it is defined):
Here's an example of solving an initial value problem EXAMPLE 5 Finding a Curve from Its Slope Function and a Point Find the curve whose slope at
In this chapter, you will explore the relationships among these problems and learn a variety of techniques for solving them 4 1 ANTIDERIVATIVES
The term indefinite integral is a synonym for antiderivative Page 2 Note: Differentiation and anti-differentiation are “inverse” operations of each other
Antiderivative Introduction Indefinite integral Integral rules Initial value problem For example, since x2 is an antiderivative of 2x, we have
Technically the indefinite integral is not a function Instead, it is a family of functions each of which is an antiderivative of f Example 7 1 8
problem this way Differentiation and antidifferentiation are reverse processes, So let's apply the initial value problem results to motion
There are several derivative anti derivative rules that you should have pretty Everyone's favorite part of math is undoubtedly the word problems
19 oct 2011 · An antiderivative of a function f on an interval I is another function F such that F/(x) = f (x) for all x ? I Examples:
Before we start looking at some examples, lets look at the process of find the antiderivative of a function The first derivative rules you learned dealt
Drill problems on derivatives and antiderivatives 1 Derivatives Find the derivative of each of the following functions (wherever it is defined): 1 f(t) = t2 + t3 − 1
Initial value problem If F and f are functions and F (x) = f(x), then F is called an antiderivative of f For example, since x2 is an antiderivative of 2x, we have ∫
Find the general antiderivatives of each of the following using you knowledge of how to find ANTIDERIVATIVES INITIAL VALUE PROBLEMS PRACTICE 1)
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Drill problems on derivatives and
antiderivatives
1 Derivatives
Find the derivative of each of the following functions (wherever it is dened):
1.f(t) =t2+t3 1t4
Answer:f0(t) = 2t3 1t2+4t5
2.y=13px+14
Answer:
dydx= 16xpx
3.f(t) = 2t3 4t2+ 3t 1. Also ndf00(t):
Answer:f0(t) = 6t2 8t+ 3; f00(t) = 12t 8
4.y=px 12
x
Answer:
dydx=12px+ ln(2)12 x
5.f(z) = ln(3)z2+ ln(4)ez
Answer:f0(z) = 2ln(3)z+ ln(4)ez
6.y=x2+ (2)x
Answer:
dydx=2x2 1+ [ln(2)](2)x
7.f() = 4p
Answer:f0() = ln(4)12p4p
1
8.f(x) =x2 px3x
Answer:f0(x) =
2x 12px!
3 x+x2 pxln(3)3x
9.f(z) =3z25z2+ 7z
Answer:f0(z) =21(5z+ 7)2
10.f(w) = (5w2+ 3)ew2
Answer:f0(w) = 10wew2+ (5w2+ 3)2wew2= (5w2+ 8)2wew2
11.f(y) =eey2
Answer:f0(y) = 2yey2eey2
12.f(z) =pz(ez+ 1)2
Answer:f0(z) =12pz1(ez+ 1)2 2pzez(ez+ 1)3
13.w(t) = (t2+ 3t)(1 e 2t)
Answer:w0(t) = 2t+ 3 +e 2t(2t2+ 4t 3)
14.f(x) =q1 cos(x)
Answer:f0(x) =sin(x)2q1 cos(x)
15.f(y) =esin(y)
Answer:f0(y) = cos(y)esin(y)
16.z=qsin(t)
Answer:
dzdt=cos(t)2qsin(t)
17.f() =2sin() + 2cos() 2sin()
Answer:f0() =2cos()
2
18.z= tan(e 3)
Answer:
dzd= 3sec2(e 3)e 3
19.y=esin(2)
Answer:
dyd=esin(2) + 2ecos(2)
20.f(y) = arcsin(y2)
Answer:f0(y) =2yp1 y4
21.f() = ln(cos())
Answer:f0() = tan()
22.f(t) = ln(ln(t)) + ln(ln(2))
Answer:f0(t) =1tln(t)
23.g(t) = arctan(3t 4)
Answer:g0(t) =31 + (3t 4)2
24.f(z) =1ln(z)
Answer:f0(z) = 1z(ln(z))2
25.f(t) = 2tet 1pt
Answer:f0(t) = 2et+ 2tet+12tpt
2 antiderivatives
Find the denite and indenite integrals below:
1. Z 3t 2t2 dt
Answer:3ln(jtj) +2t+C
3 2.Z
3cos( ) + 3q
d
Answer:3sin( ) + 2 q +C
3. Z x2+x+ 1x! dx
Answer:
12x2+x+ ln(jxj) +C
4. Z (3cos(x) 7sin(x))dx
Answer:3sin(x) + 7cos(x) +C
5.
Z1cos2(x)dx
Answer:tan(x) +C
6. Z e sin(x)cos(x)dx
Answer:esin(x)+C
7. Z =4
0(sin(t) + cos(t))dt
Answer:1
8. Z sin()(cos() + 5)7d
Answer:
18(cos() + 5)8+C 9.
Z1p4 xdx
Answer: 2p4 x+C
10. Z xe x2dx
Answer: 12e x2+C
11.
Zxcos(x2)qsin(x2)dx
Answer:
qsin(x2) +C 4
12.Zex e xex+e xdx
Answer:ln ex+e x2!
+C 13.
Z[ln(z)]2zdz
Answer:
13[ln(z)]3+C
14. Z 1
01x2+ 2x+ 1dx
Answer:
12 15. Z 0 22x+ 4x2+ 4x+ 5dx
Answer:ln(5)
16. Z tsin(t)dt
Answer: tcos(t) + sin(t) +C
17. Z yqy+ 3dy
Answer:
23(y+ 3)qy+ 335y 65
+C 18. Z t
2e5tdt
Answer:
t
2 2t5+225
e5t5+C 19. Z 1
0arctan(y)dy
Answer:
4 ln(2)2 20.
Zxp1 x2dx
Answer: p1 x2+C
5
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