Question 1

If $f(x)=\frac{x}{2}+4$ and g(x)= x + 1, how do $\int_{8}^{12} f(x) d x$ and $\int_{8}^{12} g(x) d x$ compare?

Accepted Answer

A)  $\int_{8}^{12} f(x) d x$  $\int_{8}^{12} g(x) d x$ 
C)  $\int_{8}^{12} f(x) d x$ = $\int_{8}^{12} g(x) d x$ 
A)  $\int_{8}^{12} f(x) d x$  $\int_{8}^{12} g(x) d x$ 
C)  $\int_{8}^{12} f(x) d x$ = $\int_{8}^{12} g(x) d x$ A

Question 2

The average value of a function g on 0 $\le$x $\le$2 is a constant $\bar{g}$ given by $\bar{g}=\frac{1}{2-0} \int_{0}^{2} g(d) d x$ .Also, $\int_{0}^{2}(g(x)-\bar{g})^{2} d x \geq 0$ , since $(g(x)-\bar{g})^{2}$ is a square.Which of the following must be true?

Accepted Answer

A)  $\left(\int_{0}^{2} g(x) d x\right)^{2} \leq \int_{0}^{2}(g(x))^{2} d x$ 
B)  $\left(\int_{0}^{2} g(x) d x\right)^{2} \geq \int_{0}^{2}(g(x))^{2} d x$ 
C)  $\left(\int_{0}^{2} g(x) d x\right)^{2}=\int_{0}^{2}(g(x))^{2} d x$ 
D) None of these 
A)  $\left(\int_{0}^{2} g(x) d x\right)^{2} \leq \int_{0}^{2}(g(x))^{2} d x$ 
B)  $\left(\int_{0}^{2} g(x) d x\right)^{2} \geq \int_{0}^{2}(g(x))^{2} d x$ 
C)  $\left(\int_{0}^{2} g(x) d x\right)^{2}=\int_{0}^{2}(g(x))^{2} d x$ 
D) None of these A

Question 3

The figure below shows the graph of the velocity, v, of an object (in meters/sec.).If the graph were shifted up two units, what would it mean for the motion of the object?

Accepted Answer

A) The velocity at each time would be 2 m/sec greater. 
B) The velocity at each time would be 12 m/sec greater. 
C) The velocity at each time would be 20 m/sec greater. 
D) The velocity at each time would be 120 m/sec greater. 
E) The velocity at each time would be the same. 
A) The velocity at each time would be 2 m/sec greater. 
B) The velocity at each time would be 12 m/sec greater. 
C) The velocity at each time would be 20 m/sec greater. 
D) The velocity at each time would be 120 m/sec greater. 
E) The velocity at each time would be the same. A

Question 4

At time t, in seconds, the velocity, v, in miles per hour, of a car is given by $v(t)=8+0 .8 t^{2}$ for $0 \leq t \leq 8$ .Use $\Delta$t = 2 to estimate the distance traveled during this time.Give the right-hand sum.

Accepted Answer

The answer of At time t, in seconds, the velocity,...

Question 5

How can the quantity f(a)h be represented in the picture?

Accepted Answer

A) By the slope of TV 
B) By the length of UR 
C) By the area of PQRS 
D) By the area of QRU 
A) By the slope of TV 
B) By the length of UR 
C) By the area of PQRS 
D) By the area of QRU

Question 6

The velocity of an object is given by $v(t)$ , and the acceleration is given by $a(t)$ .What is the relationship between the total change in v(t)over the interval 0 $\le$ t $\le$10 and a(t)?

Accepted Answer

A)  $\int_{0}^{10} v(t) d t=a(10)-a(0)$ 
B)  $\int_{0}^{10} a(t) d t=v(10)-v(0)$ 
C)  $\int_{0}^{10} t d t=\alpha(10)-v(10)$ 
D)  $\int_{0}^{10} v(t) d t=\int_{0}^{10} a(t) d t$ 
A)  $\int_{0}^{10} v(t) d t=a(10)-a(0)$ 
B)  $\int_{0}^{10} a(t) d t=v(10)-v(0)$ 
C)  $\int_{0}^{10} t d t=\alpha(10)-v(10)$ 
D)  $\int_{0}^{10} v(t) d t=\int_{0}^{10} a(t) d t$

Question 7

Does the quantity     $\int_{a}^{a+h} f(x) d x$ represent a length or an area in the picture?

Accepted Answer

A) An area 
B) A length 
A) An area 
B) A length

Question 8

Consider a sports car which accelerates from 0 ft/sec to 88 ft/sec in 5 seconds (88 ft/sec = 60mph).The car's velocity is given in the table below. $\begin{array}{cccccc}
t & 0 & 1 & 2 & 3 & 4 &5\
V(t) & 0 & 30 & 52 & 68 & 80&88
\end{array}$ In which time interval is the average acceleration smallest?

Accepted Answer

A) Between t = 4 and t = 5. 
B) Between t = 3 and t = 4. 
C) Between t = 2 and t = 3. 
D) Between t = 1 and t = 2. 
E) Between t = 0 and t = 1. 
A) Between t = 4 and t = 5. 
B) Between t = 3 and t = 4. 
C) Between t = 2 and t = 3. 
D) Between t = 1 and t = 2. 
E) Between t = 0 and t = 1.

Question 9

Below is the graph of the velocity, in feet per second, of a hat that is thrown up in the air from ground level.Positive velocity means upward motion. [Note that this is the graph of velocity, not distance.]   About how big is the average speed over the first 4 seconds?

Accepted Answer

A) 13.1 ft/sec 
B) 10.8 ft/sec 
C) 6.8 ft/sec 
D) 4.4 ft/sec 
A) 13.1 ft/sec 
B) 10.8 ft/sec 
C) 6.8 ft/sec 
D) 4.4 ft/sec

Question 10

What is the value of $\int_{a}^{c} f(x) d x$ if the area of A = 15 and the area of B = -4?

Accepted Answer

A) 5.5 
B) 22 
C) -11 
D) 11 
E) 19 
A) 5.5 
B) 22 
C) -11 
D) 11 
E) 19

Question 11

Consider a sports car which accelerates from 0 ft/sec to 88 ft/sec in 5 seconds (88 ft/sec = 60mph).The car's velocity is given in the table below.
 $\begin{array}{cccccc}
t & 0 & 1 & 2 & 3 & 4& 5 \
V(t) & 0 & 31 & 53 & 69 & 81& 88
\end{array}$ Find the lower bound for the distance the car travels in 5 seconds.

Accepted Answer

The answer of Consider a sports car which accelerates from...

Question 12

Two greyhound racing dogs, A and B, start at the same time and travel in the same direction along a straight track.The figure below gives the velocity, v, of each dog as a function of time t.Which dog travels the farthest?

Accepted Answer

The answer of Two greyhound racing dogs, A and B,...

Question 13

Suppose $\int_{a}^{b} g(x) d x=7$ , $\int_{a}^{b}(g(x))^{2} d x=8$ , $\int_{a}^{b} h(x) d x=1$ , and $\int_{a}^{b}(h(x))^{2} d x=4$ .Find $\int_{a}^{b} g(x) d x-\left(\int_{a}^{b}(2 h(x)) d x\right)^{2}$ .

Accepted Answer

The answer of Suppose $\int_{a}^{b} g(x) d x=7$ , \(\int_{a}^{b}(g(x))^{2}...

Question 14

Explain in words what $\int_{2}^{6} v(t) d t$ means if $v(t)$ is velocity in miles/hour and t is time, in hours.

Accepted Answer

Total change in dist

Question 15

If $\int_{0}^{5} f(x) d x=11$ and $\int_{5}^{10} f(x) d x=16$ , what is $\int_{0}^{10} f(x) d x$ ?

Accepted Answer

The answer of If $\int_{0}^{5} f(x) d x=11$ and \(\int_{5}^{10}...

Question 16

What is the value of $\int_{-a}^{a} x^{7} d x$ ?

Accepted Answer

The answer of What is the value of \(\int_{-a}^{a} x^{7}...

Question 17

Using the following figure, calculate the value of the right-hand Riemann sum f on the interval 0 $\le$ t $\le$ 12 with $\Delta$t = 6.

Accepted Answer

The answer of Using the following figure, calculate the value...

Question 18

Suppose A = the area under the curve $y=e^{-x^{2}}$ over the interval -5$\le$ x $\le$ 5.Which of the following is/are true?

Accepted Answer

A)  $A=F(5)-F(-5)$ , where $F(x)=-\frac{e^{-x^{2}}}{2 x}$ . 
B)  $1.773$$<$$A$ 
C)   $1.771<$$A<1.773$ 
D)  $1.769<$$A<1.771$ 
A)  $A=F(5)-F(-5)$ , where $F(x)=-\frac{e^{-x^{2}}}{2 x}$ . 
B)  $1.773$$<$$A$ 
C)   $1.771<$$A<1.773$ 
D)  $1.769<$$A<1.771$

Question 19

A potato is cooking in an oven.Explain in words what $\frac{1}{35-0} \int_{0}^{35} F(t) d t \approx 300$ means if $F(t)$ is the temperature of the potato, in degrees Farenheit, and t is time, in minutes.

Accepted Answer

The average temperat

Question 20

Suppose f(t)is given by the graph below.If $F(x)=\int_{0}^{x} f(t) d t$ , what is $F(4)$ ?

Accepted Answer

The answer of Suppose f(t)is given by the graph below.If...

If $f(x)=\frac{x}{2}+4$ and g(x)= x + 1, how do $\int_{8}^{12} f(x) d x$ and $\int_{8}^{12} g(x) d x$ compare?

The average value of a function g on 0 $\le$ x $\le$ 2 is a constant $\bar{g}$ given by $\bar{g}=\frac{1}{2-0} \int_{0}^{2} g(d) d x$ .Also, $\int_{0}^{2}(g(x)-\bar{g})^{2} d x \geq 0$ , since $(g(x)-\bar{g})^{2}$ is a square.Which of the following must be true?

The figure below shows the graph of the velocity, v, of an object (in meters/sec.).If the graph were shifted up two units, what would it mean for the motion of the object?

At time t, in seconds, the velocity, v, in miles per hour, of a car is given by $v(t)=8+0 .8 t^{2}$ for $0 \leq t \leq 8$ .Use $\Delta$ t = 2 to estimate the distance traveled during this time.Give the right-hand sum.

How can the quantity f(a)h be represented in the picture?

The velocity of an object is given by $v(t)$ , and the acceleration is given by $a(t)$ .What is the relationship between the total change in v(t)over the interval 0 $\le$ t $\le$ 10 and a(t)?

Consider a sports car which accelerates from 0 ft/sec to 88 ft/sec in 5 seconds (88 ft/sec = 60mph).The car's velocity is given in the table below. t 0 1 2 3 4 5 V(t) 0 30 52 68 80 88 In which time interval is the average acceleration smallest?

Below is the graph of the velocity, in feet per second, of a hat that is thrown up in the air from ground level.Positive velocity means upward motion. [Note that this is the graph of velocity, not distance.] About how big is the average speed over the first 4 seconds?

What is the value of $\int_{a}^{c} f(x) d x$ if the area of A = 15 and the area of B = -4? $What is the value of \int_{a}^{c} f(x) d x if the area of A = 15 and the area of B = -4?$

Consider a sports car which accelerates from 0 ft/sec to 88 ft/sec in 5 seconds (88 ft/sec = 60mph).The car's velocity is given in the table below. t 0 1 2 3 4 5 V(t) 0 31 53 69 81 88 Find the lower bound for the distance the car travels in 5 seconds.

Two greyhound racing dogs, A and B, start at the same time and travel in the same direction along a straight track.The figure below gives the velocity, v, of each dog as a function of time t.Which dog travels the farthest?

Suppose $\int_{a}^{b} g(x) d x=7$ , $\int_{a}^{b}(g(x))^{2} d x=8$ , $\int_{a}^{b} h(x) d x=1$ , and $\int_{a}^{b}(h(x))^{2} d x=4$ .Find $\int_{a}^{b} g(x) d x-\left(\int_{a}^{b}(2 h(x)) d x\right)^{2}$ .

Explain in words what $\int_{2}^{6} v(t) d t$ means if $v(t)$ is velocity in miles/hour and t is time, in hours.

If $\int_{0}^{5} f(x) d x=11$ and $\int_{5}^{10} f(x) d x=16$ , what is $\int_{0}^{10} f(x) d x$ ?

What is the value of $\int_{-a}^{a} x^{7} d x$ ?

Using the following figure, calculate the value of the right-hand Riemann sum f on the interval 0 $\le$ t $\le$ 12 with $\Delta$ t = 6. $Using the following figure, calculate the value of the right-hand Riemann sum f on the interval 0 \le t \le 12 with \Delta t = 6.$

Suppose A = the area under the curve $y=e^{-x^{2}}$ over the interval -5 $\le$ x $\le$ 5.Which of the following is/are true?

A potato is cooking in an oven.Explain in words what $\frac{1}{35-0} \int_{0}^{35} F(t) d t \approx 300$ means if $F(t)$ is the temperature of the potato, in degrees Farenheit, and t is time, in minutes.

Suppose f(t)is given by the graph below.If $F(x)=\int_{0}^{x} f(t) d t$ , what is $F(4)$ ? $Suppose f(t)is given by the graph below.If F(x)=\int_{0}^{x} f(t) d t , what is F(4) ?$

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 5: Key Concept- the Definite Integral

If f(x)=x2+4f(x)=\frac{x}{2}+4f(x)=2x​+4 and g(x)= x + 1, how do ∫812f(x)dx\int_{8}^{12} f(x) d x∫812​f(x)dx and ∫812g(x)dx\int_{8}^{12} g(x) d x∫812​g(x)dx compare?

The figure below shows the graph of the velocity, v, of an object (in meters/sec.).If the graph were shifted up two units, what would it mean for the motion of the object?

At time t, in seconds, the velocity, v, in miles per hour, of a car is given by v(t)=8+0.8t2v(t)=8+0 .8 t^{2}v(t)=8+0.8t2 for 0≤t≤80 \leq t \leq 80≤t≤8 .Use Δ\DeltaΔ t = 2 to estimate the distance traveled during this time.Give the right-hand sum.

How can the quantity f(a)h be represented in the picture?

The velocity of an object is given by v(t)v(t)v(t) , and the acceleration is given by a(t)a(t)a(t) .What is the relationship between the total change in v(t)over the interval 0 ≤\le≤ t ≤\le≤ 10 and a(t)?

Does the quantity ∫aa+hf(x)dx\int_{a}^{a+h} f(x) d x∫aa+h​f(x)dx represent a length or an area in the picture?

Consider a sports car which accelerates from 0 ft/sec to 88 ft/sec in 5 seconds (88 ft/sec = 60mph).The car's velocity is given in the table below. t 0 1 2 3 4 5 V(t) 0 30 52 68 80 88 In which time interval is the average acceleration smallest?

Below is the graph of the velocity, in feet per second, of a hat that is thrown up in the air from ground level.Positive velocity means upward motion. [Note that this is the graph of velocity, not distance.] About how big is the average speed over the first 4 seconds?

What is the value of ∫acf(x)dx\int_{a}^{c} f(x) d x∫ac​f(x)dx if the area of A = 15 and the area of B = -4?

Consider a sports car which accelerates from 0 ft/sec to 88 ft/sec in 5 seconds (88 ft/sec = 60mph).The car's velocity is given in the table below. t 0 1 2 3 4 5 V(t) 0 31 53 69 81 88 Find the lower bound for the distance the car travels in 5 seconds.

Two greyhound racing dogs, A and B, start at the same time and travel in the same direction along a straight track.The figure below gives the velocity, v, of each dog as a function of time t.Which dog travels the farthest?

Explain in words what ∫26v(t)dt\int_{2}^{6} v(t) d t∫26​v(t)dt means if v(t)v(t)v(t) is velocity in miles/hour and t is time, in hours.

If ∫05f(x)dx=11\int_{0}^{5} f(x) d x=11∫05​f(x)dx=11 and ∫510f(x)dx=16\int_{5}^{10} f(x) d x=16∫510​f(x)dx=16 , what is ∫010f(x)dx\int_{0}^{10} f(x) d x∫010​f(x)dx ?

What is the value of ∫−aax7dx\int_{-a}^{a} x^{7} d x∫−aa​x7dx ?

Using the following figure, calculate the value of the right-hand Riemann sum f on the interval 0 ≤\le≤ t ≤\le≤ 12 with Δ\DeltaΔ t = 6.

Suppose A = the area under the curve y=e−x2y=e^{-x^{2}}y=e−x2 over the interval -5 ≤\le≤ x ≤\le≤ 5.Which of the following is/are true?

A potato is cooking in an oven.Explain in words what 135−0∫035F(t)dt≈300\frac{1}{35-0} \int_{0}^{35} F(t) d t \approx 30035−01​∫035​F(t)dt≈300 means if F(t)F(t)F(t) is the temperature of the potato, in degrees Farenheit, and t is time, in minutes.

Suppose f(t)is given by the graph below.If F(x)=∫0xf(t)dtF(x)=\int_{0}^{x} f(t) d tF(x)=∫0x​f(t)dt , what is F(4)F(4)F(4) ?

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

If $f(x)=\frac{x}{2}+4$ and g(x)= x + 1, how do $\int_{8}^{12} f(x) d x$ and $\int_{8}^{12} g(x) d x$ compare?

At time t, in seconds, the velocity, v, in miles per hour, of a car is given by $v(t)=8+0 .8 t^{2}$ for $0 \leq t \leq 8$ .Use $\Delta$ t = 2 to estimate the distance traveled during this time.Give the right-hand sum.

The velocity of an object is given by $v(t)$ , and the acceleration is given by $a(t)$ .What is the relationship between the total change in v(t)over the interval 0 $\le$ t $\le$ 10 and a(t)?

What is the value of $\int_{a}^{c} f(x) d x$ if the area of A = 15 and the area of B = -4? $What is the value of \int_{a}^{c} f(x) d x if the area of A = 15 and the area of B = -4?$

Explain in words what $\int_{2}^{6} v(t) d t$ means if $v(t)$ is velocity in miles/hour and t is time, in hours.

If $\int_{0}^{5} f(x) d x=11$ and $\int_{5}^{10} f(x) d x=16$ , what is $\int_{0}^{10} f(x) d x$ ?

What is the value of $\int_{-a}^{a} x^{7} d x$ ?

Using the following figure, calculate the value of the right-hand Riemann sum f on the interval 0 $\le$ t $\le$ 12 with $\Delta$ t = 6. $Using the following figure, calculate the value of the right-hand Riemann sum f on the interval 0 \le t \le 12 with \Delta t = 6.$

Suppose A = the area under the curve $y=e^{-x^{2}}$ over the interval -5 $\le$ x $\le$ 5.Which of the following is/are true?

A potato is cooking in an oven.Explain in words what $\frac{1}{35-0} \int_{0}^{35} F(t) d t \approx 300$ means if $F(t)$ is the temperature of the potato, in degrees Farenheit, and t is time, in minutes.

Suppose f(t)is given by the graph below.If $F(x)=\int_{0}^{x} f(t) d t$ , what is $F(4)$ ? $Suppose f(t)is given by the graph below.If F(x)=\int_{0}^{x} f(t) d t , what is F(4) ?$