Question 1

For $H(x)=\int_{0}^{x} 6 t^{2} d t$ , find $H(1)$ .

Accepted Answer

The answer of For $H(x)=\int_{0}^{x} 6 t^{2} d t$ ,...

Question 2

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 3

Find an antiderivative F(x)with F'(x)= f (x)and F(0)= 3 when $f(x)=\sin x-\cos x$ .

Accepted Answer

The answer of Find an antiderivative F(x)with F'(x)= f (x)and...

Question 4

The area between $y=x^{5 / 2}$ , the x-axis, and x = b is approximately 79.9.Find the value of b using the Fundamental Theorem.Round to 1 decimal place.

Accepted Answer

The answer of The area between $y=x^{5 / 2}$ ,...

Question 5

A boat has constant deceleration.It was initially moving at 80 mph and stopped in a distance of 300 feet.The rate of deceleration is _____ ft/ sec².(Note: 1 mph = 22/15 ft/ sec.)Round to 2 decimal places.

Accepted Answer

The answer of A boat has constant deceleration.It was initially...

Question 6

A boulder is dropped from a cliff.A second boulder is dropped from a cliff that is half as high.How does the speed of the second boulder upon impact compare with that of the first?
A)Ths speed of the second is approximately 0.5 times the speed of the first.
B)Ths speed of the second is approximately 0.7 times the speed of the first.
C)Ths speed of the second is approximately 0.25 times the speed of the first.
D)Ths speed of the second is approximately 2 times the speed of the first.
E)Ths speed of the second is approximately the same as the speed of the first.

Accepted Answer

The speed of an object in free fall from a certain height is determined by the square root of that height (given constant acceleration due to gravity). Therefore, if the second boulder is dropped from a height that is half of the first, its speed upon impact will be the square root of 0.5 (or 1/2), which is approximately 0.7 times the speed of the first boulder.

Question 7

Use the Fundamental Theorem of Calculus to evaluate $\int_{-2}^{2}\left(9 x^{2}-6 x+5\right) d x$ .

Accepted Answer

The answer of Use the Fundamental Theorem of Calculus to...

Question 8

Evaluate $\frac{d}{d x} \int_{3^{n}}^{0} t d t$ .

Accepted Answer

The answer of Evaluate $\frac{d}{d x} \int_{3^{n}}^{0} t d t$...

Question 9

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.
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) $h(t)=4.9 t^{2}-30 t-250$

Accepted Answer

The correct formula for the height of an object in free fall is $h(t) = -\frac{1}{2}gt^2 + v_0t + h_0$, where $g$ is the acceleration due to gravity (approximately 9.8 m/s$^2$), $v_0$ is the initial velocity, and $h_0$ is the initial height. Substituting $g = 9.8$, $v_0 = -30$ (since the velocity is downward), and $h_0 = 250$, we get $h(t) = -4.9t^2 - 30t + 250$.

Question 10

Find the area of the region between $y_{1}=-2(x-4)^{2}+32$ and $y_{2}=x$ , accurate to 2 decimal places.

Accepted Answer

The answer of Find the area of the region between...

Question 11

On the moon the acceleration due to gravity is 5 feet/ sec².A brick is dropped from the top of a tower on the moon and hits the ground in 16 seconds.How many feet high is the tower?

Accepted Answer

The answer of On the moon the acceleration due to...

Question 12

The general solution of the differential equation $\frac{d y}{d x}=\frac{2}{x}+\sin 4 x$ is $2 \ln |x|+\frac{\cos 4 x}{4}+C$ .

Accepted Answer

The answer of The general solution of the differential equation...

Question 13

A boulder is dropped from a 150-foot cliff.How fast is it going when it hits the ground? Round to 2 decimal places.

Accepted Answer

The answer of A boulder is dropped from a 150-foot...

Question 14

Find $\frac{d}{d x} \int_{x}^{1}(1+t)^{3} d t$ .
A) $(1+x)^{3}$
B) $-(1+x)^{3}$
C) $\frac{1}{3}(1+x)^{4}$
D) $\frac{1}{3}(1+x)^{4}-\frac{16}{3}$

Accepted Answer

The derivative of an integral with respect to its upper limit minus the derivative with respect to its lower limit gives $-(1+x)^3$, according to the Fundamental Theorem of Calculus.

Question 15

Find $\frac{d}{d x} \int_{0}^{x}\left(\cos \left(t^{4}\right)+\sin \left(t^{4}\right)\right) d t$ .
A) $\cos \left(x^{4}\right)+\sin \left(x^{4}\right)$
B) $\sin \left(x^{4}\right)-\cos \left(x^{4}\right)$
C) $\sin \left(4 x^{3}\right)-\cos \left(4 x^{3}\right)$
D) $\frac{x^{5}}{5}\left(\sin \left(x^{4}\right)-\cos \left(x^{4}\right)\right)$

Accepted Answer

The derivative of the integral from a constant to x of a function is just the function itself evaluated at x, according to the Fundamental Theorem of Calculus.

Question 16

Write an expression for the function, f(x), with $f^{\prime}(x)=\sin ^{2} x+\sin x$ and $f(\pi / 3)=8$ .
A) $f(x)=8+\int_{0}^{x}\left(\sin ^{2} t+\sin t\right) d t$
B) $f(x)=\int_{\pi / 3}^{x}\left(\sin ^{2} t+\sin t\right) d t-8$
C) $f(x)=8+\int_{\pi / 3}^{x}\left(\sin ^{2} t+\sin t\right) d t$
D) $f(x)=8-\frac{\pi}{3}+\int_{0}^{x}\left(\sin ^{2} t+\sin t\right) d t$

Accepted Answer

Given $f^{\prime}(x)=\sin ^{2} x+\sin x$, to find $f(x)$, we integrate $f^{\prime}(x)$. The constant of integration is determined by the condition $f(\pi / 3)=8$. The correct expression is $f(x)=8+\int_{\pi / 3}^{x}\left(\sin ^{2} t+\sin t\right) d t$, which accounts for the initial condition by starting the integral at $\pi / 3$ and adding the initial value 8.

Question 17

Suppose the acceleration due to gravity on Planet A is twice that of Planet B.A brick dropped from the top of a tower on Planet A takes 20 seconds to hit the ground.A brick dropped from the top of a tower on Planet B also takes 20 seconds to hit the ground.How does the height of the tower on Planet A compare with the height of the tower on Planet B?
A)The tower on Planet A is 4 times the height of the one on Planet B.
B)The tower on Planet A is twice the height of the one on Planet B.
C)The tower on Planet A is half the height of the one on Planet B.
D)The tower on Planet A is the same as the height of the one on Planet B.

Accepted Answer

The answer of Suppose the acceleration due to gravity on...

Question 18

Evaluate $\frac{d}{d x} \int_{2}^{x} \ln \left(t^{3}\right) d t$ .

Accepted Answer

The answer of Evaluate \(\frac{d}{d x} \int_{2}^{x} \ln \left(t^{3}\right) d...

Question 19

Evaluate $\frac{d}{d x} \int_{1}^{x^{3}} \sin t d t$ .

Accepted Answer

The answer of Evaluate \(\frac{d}{d x} \int_{1}^{x^{3}} \sin t d...

Question 20

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...

Quiz 6: Constructing Antiderivatives

For $H(x)=\int_{0}^{x} 6 t^{2} d t$ , find $H(1)$ .

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

Find an antiderivative F(x)with F'(x)= f (x)and F(0)= 3 when $f(x)=\sin x-\cos x$ .

The area between $y=x^{5 / 2}$ , the x-axis, and x = b is approximately 79.9.Find the value of b using the Fundamental Theorem.Round to 1 decimal place.

A boat has constant deceleration.It was initially moving at 80 mph and stopped in a distance of 300 feet.The rate of deceleration is _____ ft/ sec².(Note: 1 mph = 22/15 ft/ sec.)Round to 2 decimal places.

A boulder is dropped from a cliff.A second boulder is dropped from a cliff that is half as high.How does the speed of the second boulder upon impact compare with that of the first?

Use the Fundamental Theorem of Calculus to evaluate $\int_{-2}^{2}\left(9 x^{2}-6 x+5\right) d x$ .

Evaluate $\frac{d}{d x} \int_{3^{n}}^{0} t d t$ .

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 area of the region between $y_{1}=-2(x-4)^{2}+32$ and $y_{2}=x$ , accurate to 2 decimal places.

On the moon the acceleration due to gravity is 5 feet/ sec².A brick is dropped from the top of a tower on the moon and hits the ground in 16 seconds.How many feet high is the tower?

The general solution of the differential equation $\frac{d y}{d x}=\frac{2}{x}+\sin 4 x$ is $2 \ln |x|+\frac{\cos 4 x}{4}+C$ .

A boulder is dropped from a 150-foot cliff.How fast is it going when it hits the ground? Round to 2 decimal places.

Find $\frac{d}{d x} \int_{x}^{1}(1+t) ^{3} d t$ .

Find $\frac{d}{d x} \int_{0}^{x}\left(\cos \left(t^{4}\right) +\sin \left(t^{4}\right) \right) d t$ .

Write an expression for the function, f(x) , with $f^{\prime}(x) =\sin ^{2} x+\sin x$ and $f(\pi / 3) =8$ .

Evaluate $\frac{d}{d x} \int_{2}^{x} \ln \left(t^{3}\right) d t$ .

Evaluate $\frac{d}{d x} \int_{1}^{x^{3}} \sin t d t$ .

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

Quiz 6: Constructing Antiderivatives

For H(x)=∫0x6t2dtH(x)=\int_{0}^{x} 6 t^{2} d tH(x)=∫0x​6t2dt , find H(1)H(1)H(1) .

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 F(x)with F'(x)= f (x)and F(0)= 3 when f(x)=sin⁡x−cos⁡xf(x)=\sin x-\cos xf(x)=sinx−cosx .

The area between y=x5/2y=x^{5 / 2}y=x5/2 , the x-axis, and x = b is approximately 79.9.Find the value of b using the Fundamental Theorem.Round to 1 decimal place.

A boat has constant deceleration.It was initially moving at 80 mph and stopped in a distance of 300 feet.The rate of deceleration is _____ ft/ sec2.(Note: 1 mph = 22/15 ft/ sec.)Round to 2 decimal places.

A boulder is dropped from a cliff.A second boulder is dropped from a cliff that is half as high.How does the speed of the second boulder upon impact compare with that of the first?

Use the Fundamental Theorem of Calculus to evaluate ∫−22(9x2−6x+5)dx\int_{-2}^{2}\left(9 x^{2}-6 x+5\right) d x∫−22​(9x2−6x+5)dx .

Evaluate ddx∫3n0tdt\frac{d}{d x} \int_{3^{n}}^{0} t d tdxd​∫3n0​tdt .

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 area of the region between y1=−2(x−4)2+32y_{1}=-2(x-4)^{2}+32y1​=−2(x−4)2+32 and y2=xy_{2}=xy2​=x , accurate to 2 decimal places.

On the moon the acceleration due to gravity is 5 feet/ sec2.A brick is dropped from the top of a tower on the moon and hits the ground in 16 seconds.How many feet high is the tower?

The general solution of the differential equation dydx=2x+sin⁡4x\frac{d y}{d x}=\frac{2}{x}+\sin 4 xdxdy​=x2​+sin4x is 2ln⁡∣x∣+cos⁡4x4+C2 \ln |x|+\frac{\cos 4 x}{4}+C2ln∣x∣+4cos4x​+C .

A boulder is dropped from a 150-foot cliff.How fast is it going when it hits the ground? Round to 2 decimal places.

Find ddx∫x1(1+t) 3dt\frac{d}{d x} \int_{x}^{1}(1+t) ^{3} d tdxd​∫x1​(1+t) 3dt .

Find ddx∫0x(cos⁡(t4) +sin⁡(t4) ) dt\frac{d}{d x} \int_{0}^{x}\left(\cos \left(t^{4}\right) +\sin \left(t^{4}\right) \right) d tdxd​∫0x​(cos(t4) +sin(t4) ) dt .

Write an expression for the function, f(x) , with f′(x) =sin⁡2x+sin⁡xf^{\prime}(x) =\sin ^{2} x+\sin xf′(x) =sin2x+sinx and f(π/3) =8f(\pi / 3) =8f(π/3) =8 .

Evaluate ddx∫2xln⁡(t3)dt\frac{d}{d x} \int_{2}^{x} \ln \left(t^{3}\right) d tdxd​∫2x​ln(t3)dt .

Evaluate ddx∫1x3sin⁡tdt\frac{d}{d x} \int_{1}^{x^{3}} \sin t d tdxd​∫1x3​sintdt .

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

For $H(x)=\int_{0}^{x} 6 t^{2} d t$ , find $H(1)$ .

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

Find an antiderivative F(x)with F'(x)= f (x)and F(0)= 3 when $f(x)=\sin x-\cos x$ .

The area between $y=x^{5 / 2}$ , the x-axis, and x = b is approximately 79.9.Find the value of b using the Fundamental Theorem.Round to 1 decimal place.

A boat has constant deceleration.It was initially moving at 80 mph and stopped in a distance of 300 feet.The rate of deceleration is _____ ft/ sec².(Note: 1 mph = 22/15 ft/ sec.)Round to 2 decimal places.

Use the Fundamental Theorem of Calculus to evaluate $\int_{-2}^{2}\left(9 x^{2}-6 x+5\right) d x$ .

Evaluate $\frac{d}{d x} \int_{3^{n}}^{0} t d t$ .

Find the area of the region between $y_{1}=-2(x-4)^{2}+32$ and $y_{2}=x$ , accurate to 2 decimal places.

On the moon the acceleration due to gravity is 5 feet/ sec².A brick is dropped from the top of a tower on the moon and hits the ground in 16 seconds.How many feet high is the tower?

The general solution of the differential equation $\frac{d y}{d x}=\frac{2}{x}+\sin 4 x$ is $2 \ln |x|+\frac{\cos 4 x}{4}+C$ .

Find $\frac{d}{d x} \int_{x}^{1}(1+t) ^{3} d t$ .

Find $\frac{d}{d x} \int_{0}^{x}\left(\cos \left(t^{4}\right) +\sin \left(t^{4}\right) \right) d t$ .

Write an expression for the function, f(x) , with $f^{\prime}(x) =\sin ^{2} x+\sin x$ and $f(\pi / 3) =8$ .

Evaluate $\frac{d}{d x} \int_{2}^{x} \ln \left(t^{3}\right) d t$ .

Evaluate $\frac{d}{d x} \int_{1}^{x^{3}} \sin t d t$ .

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