ap calc bc 2023_在线真题试卷与模拟练习_ap calc bc 2023

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If $$g(x)=x^{2}\sqrt{x+1}$$, then $$g'(3)=$$

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An airplane's height above sea level is modeled by the differentiable function $$y=h(t)$$, where $$t$$ is the number of minutes after takeoff and $$h(t)$$ is measured in feet. The graph of $$h$$ is shown in the figure for $$0\leq t\leq15$$. Of the following, which is the best estimate of the rate at which the plane is rising above sea level, in feet per minute, at time $$t=9$$?

Let $$f$$ be a vector-valued function with $$f(0)=\langle-2,3\rangle$$. If the instantaneous rate of change of $$f$$ with respect to $$t$$ is given by $$\langle3t^{2}+4t+1,\ 2t^{3}+t-2\rangle$$, what is $$f(1)$$?

$$\lim_{x\to4}\frac{2-\sqrt{x}}{x-4}$$

Let $$y=f(x)$$ be the solution to the differential equation $$\frac{dy}{dx}=x+y^{2}$$ with initial condition $$f(1)=2$$. What is the approximation for $$f(1.4)$$, if Euler's method is used, starting at $$x=1$$ with two steps of equal size?

（含图）

A portion of the slope field for the differential equation $$\frac{dy}{dx}=x-y$$ is shown in the figure. If the linear function $$y=f(x)$$ is the solution to the differential equation such that $$f(-1)=-2$$, then $$f(3)=$$

If $$f(x)=\sin(e^{3x})$$, then $$f'(x)=$$

If the series $$\sum_{n=0}^{\infty}(-1)^{n}\frac{B^{n+1}}{A^{n}}$$ converges, where $$A\neq0$$ and $$B\neq0$$, what is the sum of the series?

Of the following series, which can be used with the comparison test to show that $$\sum_{n=3}^{\infty}\frac{\ln n}{n}$$ diverges?

A particle moves in the $$xy$$-plane so that its position at any time $$t$$ is given by the parametric equations $$x(t)=t^{2}-t^{3}$$ and $$y(t)=9-6t$$. What is the speed of the particle at time $$t=2$$?

Which of the following is the solution to the differential equation $$\frac{dy}{dx}=-3y$$ with the initial condition $$y(0)=7$$?

Consider the series $$\sum_{n=1}^{\infty}\frac{(-1)^{n}}{\cos(\frac{1}{n})}$$. Which of the following statements is true?

If $$f$$ is a function such that $$\lim_{x \to \infty} f(x) = 0$$, which of the following could be an expression for $$f(x)$$?

$$\int \frac{x+1}{x(x+3)} dx =$$

$$f(x) = \begin{cases} x^2 - 2x, & x \leq 4 \\ g(x), & x > 4 \end{cases}$$
Let $$f$$ be the function defined as shown. If $$f$$ is differentiable, which of the following could define $$g(x)$$?

The function $$y = f(x)$$ satisfies the differential equation $$\frac{dy}{dx} = 2xy$$ with the initial condition $$f(0) = 5$$. What is the value of $$f(2)$$?

Which of the following gives the length of the curve $$y = \tan x$$ from $$x = -\frac{\pi}{4}$$ to $$x = \frac{\pi}{4}$$?

If $$y = f(x)$$ is a function such that $$\frac{dy}{dx} = \frac{2x - y}{x}$$, then $$\frac{d^2y}{dx^2} =$$

Which of the following series converges to $$\frac{1}{e}$$?

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