Question 1

Evaluate $\frac{d}{d x} \int_{\sin (x)}^{4} \tan (t) d t$ .

Accepted Answer

The answer of Evaluate \(\frac{d}{d x} \int_{\sin (x)}^{4} \tan (t)...

Question 2

Find an antiderivative of $e^{t}+e^{7}$ .

Accepted Answer

A)  $\frac{e^{t+1}}{t+1}+e^{7} t+C$ 
B)  $\frac{e^{t}}{t}+e^{7} t+C$ 
C)  $e^{t}+C$ 
D)  $e^{t}+e^{7} t+C$ 
A)  $\frac{e^{t+1}}{t+1}+e^{7} t+C$ 
B)  $\frac{e^{t}}{t}+e^{7} t+C$ 
C)  $e^{t}+C$ 
D)  $e^{t}+e^{7} t+C$

Question 3

Evaluate $\frac{d}{d x} \int_{e}^{x} \log _{11}\left(t^{21}\right) \sin (\sqrt{t}) d t$ .

Accepted Answer

The answer of Evaluate \(\frac{d}{d x} \int_{e}^{x} \log _{11}\left(t^{21}\right) \sin...

Question 4

Find the value of G( $\pi$/2)where G '(x)= 2 sin x cos x and G(0)= 1.

Accepted Answer

The answer of Find the value of G( $\pi$/2)where G...

Question 5

Find an antiderivative of $\pi+x^{6}+\frac{1}{\pi x^{6}}$ .

Accepted Answer

A)  $\pi x+\frac{x^{7}}{7}-\frac{1}{5 \pi x^{5}}+C$ 
B)  $\pi x+\frac{x^{7}}{7}+\frac{1}{7 \pi x^{7}}+C$ 
C)  $\pi x+\frac{x^{7}}{7}+\frac{7}{\pi x^{7}}+C$ 
D)  $\frac{\pi^{2}}{2}+\frac{x^{7}}{7}-\frac{7}{\pi x^{7}}+C$ 
A)  $\pi x+\frac{x^{7}}{7}-\frac{1}{5 \pi x^{5}}+C$ 
B)  $\pi x+\frac{x^{7}}{7}+\frac{1}{7 \pi x^{7}}+C$ 
C)  $\pi x+\frac{x^{7}}{7}+\frac{7}{\pi x^{7}}+C$ 
D)  $\frac{\pi^{2}}{2}+\frac{x^{7}}{7}-\frac{7}{\pi x^{7}}+C$

Question 6

At time t = 0, a bowling ball rolls off a 250-meter ledge with velocity 30 meters/sec downward.Express its height, h(t), in meters above the ground as a function of time, t, in seconds.

Accepted Answer

A)  $h(t)=-4.9 t^{2}-30 t+250$ 
B)  $h(t)=-4.9 t^{2}-30 t-250$ 
C)  $h(t)=4.9 t^{2}-30 t+250$ 
D)  $h(t)=4.9 t^{2}-30 t-250$ 
A)  $h(t)=-4.9 t^{2}-30 t+250$ 
B)  $h(t)=-4.9 t^{2}-30 t-250$ 
C)  $h(t)=4.9 t^{2}-30 t+250$ 
D)  $h(t)=4.9 t^{2}-30 t-250$

Question 7

Find the solution of the initial value problems $\frac{d K}{d t}=2-\cos 5 t$ when $K(0)=-12$ .

Accepted Answer

A)  $2 t-\frac{\sin 5 t}{5}-12$ 
B)  $2 t-\frac{\sin 5 t}{5}+12$ 
C)  $2 t+\frac{\sin 5 t}{5}-12$ 
D)  $2 t+\frac{\sin 5 t}{5}+12$ 
A)  $2 t-\frac{\sin 5 t}{5}-12$ 
B)  $2 t-\frac{\sin 5 t}{5}+12$ 
C)  $2 t+\frac{\sin 5 t}{5}-12$ 
D)  $2 t+\frac{\sin 5 t}{5}+12$

Question 8

For $-2 \leq x \leq 2$ , define $F(x)=\int_{-2}^{x} \sqrt{4-t^{2}} d t$ .What is the value of $F(0)$ ? Round to 2 decimal places.

Accepted Answer

The answer of For $-2 \leq x \leq 2$ ,...

Question 9

A factory is dumping pollutants into a lake continuously at the rate of $\frac{t^{2 / 3}}{40}$ tons per week, where t is the time in weeks since the factory commenced operations.After one year of operation, how many tons of pollutant has the factory dumped into the lake? Round to 2 decimal places.

Accepted Answer

The answer of A factory is dumping pollutants into a...

Question 10

A ball is thrown vertically upwards from the top of a 256-foot cliff with initial velocity of 96 feet per second.Find its maximum height (in ft).

Accepted Answer

The answer of A ball is thrown vertically upwards from...

Evaluate $\frac{d}{d x} \int_{\sin (x)}^{4} \tan (t) d t$ .

Find an antiderivative of $e^{t}+e^{7}$ .

Evaluate $\frac{d}{d x} \int_{e}^{x} \log _{11}\left(t^{21}\right) \sin (\sqrt{t}) d t$ .

Find the value of G( $\pi$ /2)where G '(x)= 2 sin x cos x and G(0)= 1.

Find an antiderivative of $\pi+x^{6}+\frac{1}{\pi x^{6}}$ .

At time t = 0, a bowling ball rolls off a 250-meter ledge with velocity 30 meters/sec downward.Express its height, h(t), in meters above the ground as a function of time, t, in seconds.

Find the solution of the initial value problems $\frac{d K}{d t}=2-\cos 5 t$ when $K(0)=-12$ .

For $-2 \leq x \leq 2$ , define $F(x)=\int_{-2}^{x} \sqrt{4-t^{2}} d t$ .What is the value of $F(0)$ ? Round to 2 decimal places.

A factory is dumping pollutants into a lake continuously at the rate of $\frac{t^{2 / 3}}{40}$ tons per week, where t is the time in weeks since the factory commenced operations.After one year of operation, how many tons of pollutant has the factory dumped into the lake? Round to 2 decimal places.

A ball is thrown vertically upwards from the top of a 256-foot cliff with initial velocity of 96 feet per second.Find its maximum height (in ft).

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

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 6: Constructing Antiderivatives

Evaluate ddx∫sin⁡(x)4tan⁡(t)dt\frac{d}{d x} \int_{\sin (x)}^{4} \tan (t) d tdxd​∫sin(x)4​tan(t)dt .

Find an antiderivative of et+e7e^{t}+e^{7}et+e7 .

Evaluate ddx∫exlog⁡11(t21)sin⁡(t)dt\frac{d}{d x} \int_{e}^{x} \log _{11}\left(t^{21}\right) \sin (\sqrt{t}) d tdxd​∫ex​log11​(t21)sin(t​)dt .

Find the value of G( π\piπ /2)where G '(x)= 2 sin x cos x and G(0)= 1.

Find an antiderivative of π+x6+1πx6\pi+x^{6}+\frac{1}{\pi x^{6}}π+x6+πx61​ .

At time t = 0, a bowling ball rolls off a 250-meter ledge with velocity 30 meters/sec downward.Express its height, h(t), in meters above the ground as a function of time, t, in seconds.

Find the solution of the initial value problems dKdt=2−cos⁡5t\frac{d K}{d t}=2-\cos 5 tdtdK​=2−cos5t when K(0)=−12K(0)=-12K(0)=−12 .

For −2≤x≤2-2 \leq x \leq 2−2≤x≤2 , define F(x)=∫−2x4−t2dtF(x)=\int_{-2}^{x} \sqrt{4-t^{2}} d tF(x)=∫−2x​4−t2​dt .What is the value of F(0)F(0)F(0) ? Round to 2 decimal places.

A ball is thrown vertically upwards from the top of a 256-foot cliff with initial velocity of 96 feet per second.Find its maximum height (in ft).

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

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

Evaluate $\frac{d}{d x} \int_{\sin (x)}^{4} \tan (t) d t$ .

Find an antiderivative of $e^{t}+e^{7}$ .

Evaluate $\frac{d}{d x} \int_{e}^{x} \log _{11}\left(t^{21}\right) \sin (\sqrt{t}) d t$ .

Find the value of G( $\pi$ /2)where G '(x)= 2 sin x cos x and G(0)= 1.

Find an antiderivative of $\pi+x^{6}+\frac{1}{\pi x^{6}}$ .

Find the solution of the initial value problems $\frac{d K}{d t}=2-\cos 5 t$ when $K(0)=-12$ .

For $-2 \leq x \leq 2$ , define $F(x)=\int_{-2}^{x} \sqrt{4-t^{2}} d t$ .What is the value of $F(0)$ ? Round to 2 decimal places.