BASIC CALCULUS REVIEWER Power Rule
-used to integrate the terms that are of the form
"variable raised to exponent”
∫ f ( x ) dx=F ( x ) +C
f = integrand EXAMPLE: Except
∫ √ x dx ∫∫
3
dx = variable integration 7 2 5 dx 5
C = Arbitrary Constant x dx
Cuz there is no variable raised to
∫ x −3 dx exponent
constant
and there is only a
so we can use
(which is 5 )
constant multiple rule .
Transcendental Functions
- defined as functions that cannot be written using algebraic
operations of addition, multiplication, or their inverse
operation
What would you do first to integrate the
function?
∫ 4√ x 7 dx
Which statement is True?
- A function F is an antiderivative of the function f on an
7/4
interval I if F’(x)=f(x) for all values of x in l
Evaluate the Integral
1. ∫ ( 1−6 x 2−8 x ) dx = x−2 x 3−4 x 2 +C
5 3
4 2 10 2
2. ∫ √ x ( 2 x−5 ) dx = x − x +C
5 3
Word Problem
Allan is explaining the wonders of nature to his daughter.
−1 He and his daughter went to the rooftop of a 120m-high
3. ∫ (x−4 −4)dx = 3 x 3 −4 x+C building, then he threw a stone from the rooftop.
2 5 3 1 1. Neglecting the air friction, at what time will the stone
4. ∫ t −2t
√t
+1
dt
2
5
4
= t 2 − t 2 +2 t 2 +C
3
fall to the ground?
a. 3.95s
b. 4.55s
1 1 c. 4.95s
5. ∫ 2 e x dx = 2 e x +C d. 5.55s
2. With what velocity will a stone strike the ground if
sinx −cosx dropped from the top of a building?
6. ∫ 3 dx =
3
+C a. -48.51 m/s
b. -9.8 m/s
7. ∫ 2 se c 2 xdx = 2 tan x+ C c. 9.8 m/s
d. 48.51 m/s
dx
8. ∫ = sin −1 x +C Rate of Change
√ 1−x 2
The rate of change dy/dt in the population y with
7 respect to the time t is given by dy/dt = ky, where k
9. ∫ ¿ ¿ = 23 x 6−2 x−4 +25 x 5
+C = b - d. Which of the following statements is TRUE?
a. If k is positive, that is when b > d, then there are more
−1 6
10. ∫ x ( 3−x ) 2 5
=
12
( 3−x2 ) +C deaths than births and dy/dt denotes decay
b. If k is negative, that is when b < d, then there are more
−1 deaths than births, and dy/dt denotes growth
∫ xsin ( 2 x2 ) dx = 4 cos ( 2 x ) +C
2
11.
c. If k is positive, that is when b > d, then there are more
12. ∫ sec ( 2 x +1 ) tan ( 2 x +1 ) dx = births than deaths and dy/dt denotes growth
d. If k is negative, that is when b < d, then there are more
1
sec ( 2 x+1 )+C births than deaths and dy/dt denotes decay
2
∫ t 2 ( 12 t 3−4 ) dt = 29 ( 12 t3 −4 ) +C
2 3
13.
Law of Exponential Growth
1. An experimental population of fruit flies increases
2
according to the law of exponential growth. There
−3 x −3 x
2
were 100 flies after the second day of the
-used to integrate the terms that are of the form
"variable raised to exponent”
∫ f ( x ) dx=F ( x ) +C
f = integrand EXAMPLE: Except
∫ √ x dx ∫∫
3
dx = variable integration 7 2 5 dx 5
C = Arbitrary Constant x dx
Cuz there is no variable raised to
∫ x −3 dx exponent
constant
and there is only a
so we can use
(which is 5 )
constant multiple rule .
Transcendental Functions
- defined as functions that cannot be written using algebraic
operations of addition, multiplication, or their inverse
operation
What would you do first to integrate the
function?
∫ 4√ x 7 dx
Which statement is True?
- A function F is an antiderivative of the function f on an
7/4
interval I if F’(x)=f(x) for all values of x in l
Evaluate the Integral
1. ∫ ( 1−6 x 2−8 x ) dx = x−2 x 3−4 x 2 +C
5 3
4 2 10 2
2. ∫ √ x ( 2 x−5 ) dx = x − x +C
5 3
Word Problem
Allan is explaining the wonders of nature to his daughter.
−1 He and his daughter went to the rooftop of a 120m-high
3. ∫ (x−4 −4)dx = 3 x 3 −4 x+C building, then he threw a stone from the rooftop.
2 5 3 1 1. Neglecting the air friction, at what time will the stone
4. ∫ t −2t
√t
+1
dt
2
5
4
= t 2 − t 2 +2 t 2 +C
3
fall to the ground?
a. 3.95s
b. 4.55s
1 1 c. 4.95s
5. ∫ 2 e x dx = 2 e x +C d. 5.55s
2. With what velocity will a stone strike the ground if
sinx −cosx dropped from the top of a building?
6. ∫ 3 dx =
3
+C a. -48.51 m/s
b. -9.8 m/s
7. ∫ 2 se c 2 xdx = 2 tan x+ C c. 9.8 m/s
d. 48.51 m/s
dx
8. ∫ = sin −1 x +C Rate of Change
√ 1−x 2
The rate of change dy/dt in the population y with
7 respect to the time t is given by dy/dt = ky, where k
9. ∫ ¿ ¿ = 23 x 6−2 x−4 +25 x 5
+C = b - d. Which of the following statements is TRUE?
a. If k is positive, that is when b > d, then there are more
−1 6
10. ∫ x ( 3−x ) 2 5
=
12
( 3−x2 ) +C deaths than births and dy/dt denotes decay
b. If k is negative, that is when b < d, then there are more
−1 deaths than births, and dy/dt denotes growth
∫ xsin ( 2 x2 ) dx = 4 cos ( 2 x ) +C
2
11.
c. If k is positive, that is when b > d, then there are more
12. ∫ sec ( 2 x +1 ) tan ( 2 x +1 ) dx = births than deaths and dy/dt denotes growth
d. If k is negative, that is when b < d, then there are more
1
sec ( 2 x+1 )+C births than deaths and dy/dt denotes decay
2
∫ t 2 ( 12 t 3−4 ) dt = 29 ( 12 t3 −4 ) +C
2 3
13.
Law of Exponential Growth
1. An experimental population of fruit flies increases
2
according to the law of exponential growth. There
−3 x −3 x
2
were 100 flies after the second day of the
