The function $$F$$ is defined by $$F(x) = \int \cos(x) dx$$. If $$F(0)=0$$, what is the value of $$F(\pi/2)$$?
Solving for functions given an antiderivative and initial condition (motion included).
28 questions
AP Learning Target(s): LT 8: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
The function $$F$$ is defined by $$F(x) = \int \cos(x) dx$$. If $$F(0)=0$$, what is the value of $$F(\pi/2)$$?
$$\sin(0)$$
$$1$$
$$\frac{\pi}{2}$$
$$0$$
The derivative of a function $$f$$ is given by $$f'(x) = 3x^2 - 2x + 5$$. If $$f(0) = 4$$, what is the value of $$f(2)$$?
18
12
20
14
Let $$g$$ be the antiderivative of $$f(x) = 3x^2 - 4x + 5$$ such that $$g(1)=10$$. What is $$g(x)$$?
$$x^3 - 2x^2 + 5x + 10$$
$$x^3 - 2x^2 + 5x + 4$$
$$x^3 - 4x^2 + 5x + 10$$
$$x^3 - 2x^2 + 5x + 6$$
The growth rate of a bacterial culture is modeled by the equation $$\frac{dN}{dt}=0.5t^2$$, where $N$ is the number of bacteria and $t$ is the time in hours. If the culture contains 100 bacteria at time $t=0$, what is the number of bacteria at time $t=4$?
100
$$\frac{332}{3}$$
110
108
Let $$H(x)$$ be an antiderivative of $$\displaystyle \frac{x^3 + \sin x}{x^2 + 2}$$. If $$H(5) = \pi$$, what is the value of $$H(2)$$?
$$4.626$$
$$-9.008$$
$$-5.867$$
$$12.150$$
The acceleration of a particle moving along the x-axis is given by $$a(t)=12t^2$$ for time $$t geq 0$$. The position of the particle is 1 at time $$t=0$$ and 7 at time $$t=1$$. What is the velocity of the particle at time $$t=0$$?
5
3
6
4
A particle moves along the x-axis. For time $$t \ge 0$$, the acceleration of the particle is given by $$a(t)=6\cos(t)$$. The position of the particle is 2 at time $$t=0$$, and the position is 8 at time $$t=\frac{\pi}{2}$$. What is the velocity of the particle at time $$t=0$$?
-4
4
0
6
The acceleration of a particle moving along the x-axis is given by $$a(t)=8t+6$$ for time $$t geq 0$$. The position of the particle is 7 at time $$t=0$$ and 18 at time $$t=1$$. What is the velocity of the particle at time $$t=0$$?
$$\frac{19}{3}$$
$$\frac{22}{3}$$
$$\frac{18}{3}$$
$$\frac{20}{3}$$
The acceleration of a particle moving along the x-axis is given by $$a(t)=8t+2$$ for time $$t \ge 0$$. The particle is at position $$x=2$$ at time $$t=0$$ and at position $$x=9$$ at time $$t=1$$. What is the velocity of the particle at time $$t=0$$?
$$\frac{16}{3}$$
$$\frac{12}{3}$$
$$\frac{14}{3}$$
$$\frac{13}{3}$$
For time t≥0, the acceleration of a particle moving along the x-axis is given by $$a(t)=12t$$. If the particle's position is 4 at time t=0 and 12 at time t=1, what is the velocity of the particle at time t=0?
6
2
4
8
The acceleration of a particle moving along the x-axis is given by $$a(t) = 6t$$ for time $$t \geq 0$$. The position of the particle is 1 at time $$t=0$$ and 15 at time $$t=2$$. What is the velocity of the particle at time $$t=0$$?
1
3
2
4
A particle moves along the x-axis. For time $$t \ge 0$$, the acceleration of the particle is given by $$a(t) = 6t^2 - 4$$. The position of the particle is 1 at $$t=0$$ and 5 at $$t=1$$. What is the velocity of the particle at $$t=0$$?
$$\frac{13}{2}$$
$$\frac{11}{2}$$
$$\frac{10}{2}$$
$$\frac{9}{2}$$
A particle moves along the x-axis such that its acceleration at time $$t \ge 0$$ is given by $$a(t) = 6\sqrt{t}$$. The position of the particle is 2 at time $$t=0$$ and 15 at time $$t=1$$. What is the velocity of the particle at time $$t=0$$?
$$\frac{58}{5}$$
$$\frac{55}{5}$$
$$\frac{57}{5}$$
$$\frac{56}{5}$$
For a particle moving along the x-axis, the acceleration at time $$t \geq 0$$ is given by $$a(t) = 3t^2$$. The position of the particle is 2 at time $$t=0$$ and 8 at time $$t=1$$. What is the velocity of the particle at time $$t=0$$?
$$\frac{25}{4}$$
$$\frac{23}{4}$$
$$\frac{22}{4}$$
$$\frac{21}{4}$$
A particle moves along the x-axis with an acceleration function `a(t) = 6t - 4` for `0 ≤ t ≤ 4`. At time `t = 0`, the particle's velocity is `v(0) = -3` and its position is `x(0) = 2`. Which of the following expressions represents the position of the particle at time `t = 4`?
2 + $\displaystyle\int_{0}^{4}$ [-3 + (6t-4)t]dt
2 + (-3)(4) + ∑ᵢ₌₁⁴ (6tᵢ-4)Δt
2 + $\displaystyle\int_{0}^{4}$ [-3 + $\displaystyle\int_{0}$ᵗ (6s-4)ds]dt
2 - 3(4) + $\displaystyle\int_{0}^{4}$ (6t-4)dt
The acceleration of a car is given by $$a(t) = 4t - 6$$ meters per second squared. If the car's initial velocity is $$v(0) = 2$$ meters per second, what is the velocity of the car at time $$t = 3$$ seconds?
8 m/s
4 m/s
2 m/s
5 m/s
The derivative of a function $$h$$ is given by $$h'(x) = \frac{e^x}{(1+e^x)^2}$$. If $$h(0) = 5$$, what is the value of $$h(\ln 3)$$?
5.5
5.25
5.75
6
A particle moves along the x-axis with an acceleration given by $$a(t)=2t+4$$ for time $$t \ge 0$$. If the particle is at position $$x=5$$ at time $$t=0$$ and at position $$x=12$$ at time $$t=1$$, what is the velocity of the particle at time $$t=0$$?
$$\frac{15}{3}$$
$$\frac{13}{3}$$
$$\frac{11}{3}$$
$$\frac{14}{3}$$
Let `g(t)` be the rate of change of a particle's position at time `t`. The position of the particle is `s = 3` at time `t = 2`. Which of the following gives the position function `s(x)` for `x ≥ 2`?
s(x) = g(x) + 3
s(x) = 3 + $\displaystyle\int_{2}$ˣ g(t) dt
s(x) = $\displaystyle\int_{0}$ˣ g(t) dt + 3
s(x) = 3 - $\displaystyle\int_{2}$ˣ g(t) dt
A particle moves along a straight line with velocity $$v(t) = t^2 - 4t + 3$$ for $$0 \leq t \leq 5$$. If the particle's initial position is $$s(0) = -2$$, what is its position at time $$t = 5$$?
s(5) = 8/3
s(5) = -2/3
s(5) = 14/3
s(5) = -8/3
What is the solution to the initial value problem $$\frac{dy}{dx} = 3x^2$$ with the initial condition $$y(1)=5$$?
$$x^3+4$$
$$x^3+5$$
$$3*x^2+4$$
$$x^3+1$$
If $$f'(x) = 3x^2 + 2x$$ and $$f(2) = 3$$, what is the value of $$f(1)$$?
$$10$$
$$-10$$
$$13$$
$$-7$$
The derivative of a function $$f$$ is given by $$f'(x) = \cos(x^2)$$. If $$f(3) = 7$$, what is the value of $$f(2)$$?
$$0.241$$
$$5.831$$
$$6.759$$
$$6.416$$
Let $$g$$ be a function whose first derivative is given by $$g'(x) = \sqrt{x^3 + x}$$ for $$x > 0$$. Given that $$g(2) = -7$$, what is the value of $$g(5)$$?
11.402
20.899
13.899
4.402
If $$f'(x) = \displaystyle\frac{2}{x}$$ and $$f(\sqrt{e}) = 5$$, what is the value of $$f(e)$$?
$$\displaystyle 25$$
$$\displaystyle 2$$
$$\displaystyle 6$$
$$\displaystyle \ln 25$$
The function F is defined such that F'(x) = 4x³ - 2x + 7 and F(1) = 10. What is F(x)?
x⁴ - x² + 7x + 10
x⁴ - x² + 7x
x⁴ - x² + 7x + 3
x⁴ - x² + 7x + C
Let $$g$$ be a function for which $$g'(x) = x\sqrt{1-x^2}$$ and $$g(0) = 3$$. What is the value of $$g\left(\frac{1}{2}\right)$$?
$$\frac{10}{3} - \frac{\sqrt{3}}{8}$$
3 - 1/3
3 - 1/12
3 + 1/6
Which of the following is an example of correctly using an integral to calculate the final velocity of a car, given its acceleration function $$a(t)=6*t$$ and an initial velocity of 10 m/s over the interval $$0\le t\le 3$$?
Final velocity = $$10+\frac{6*3^2}{2}=10+27=37 \text{ m/s}$$, though the method is unclear.
Final velocity = $$10+6*3=28 \text{ m/s}$$, by multiplying acceleration directly by time.
Final velocity = $$10+\int_{0}^{3}6*t dt=10+27=37 \text{ m/s}$$
Final velocity = $$\int_{0}^{3}6*t dt=27 \text{ m/s}$$, neglecting the initial velocity.