\(\operatorname*{lim}\limits_{x \rightarrow \dfrac{\pi}{3}} \dfrac{\cos x - \dfrac{1}{2}}{x - \dfrac{\pi}{3}}\) is

If \(f(x) = \sec(2 x)\), then \(f'(x) =\)

\(\displaystyle\int_{0}^{1} x(\sqrt{x} + 1) d x =\)

What is the slope of the tangent line to the graph of \(\dfrac{x y + 1}{y + 2} = 1\) at point \((2, 1)\)?

What is the x-coordinate of the point of inflection for the graph of \(y = x^3 + 3 x^2 - 1\)?

\(\displaystyle\int_{0}^{x} \sin t \cos^2 t d t =\)

A particle moves along the x-axis so that its position at time \(t > 0\) is given by \(x(t) = 3 \ln t - 6 t + 7\). At what time \(t > 0\) is the velocity of the particle equal to zero?

If \(F(x) = \displaystyle\int_{1}^{\ln x} \sqrt{\cos t} d t\), then \(F'(x) =\)

The function \(f\) is differentiable on the closed interval \([1, 4]\), and it has values as follows: \(f(1) = 3\), \(f(2) = k\), and \(f(4) = 5 k + 2\). For which of the following values of \(k\) must there exist two points \(a\) and \(b\) on the open interval \((1, 4)\) with \(f'(a) = f'(b)\)?

Which of the following is a solution to the differential equation \(\dfrac{d y}{d x} = \dfrac{\sec^2 x}{y^2}\)?

If \(g(x) = f(x^2)\), what is \(g''(x)\)?

What's the instantaneous rate of change of \(y = \ln(\cos x)\) at the point \(x = \dfrac{\pi}{6}\)?

For what values of \(x\) is the function \(f(x) = 1 + x^2 - 2 x^4\) increasing?

Which of the following is an equation of the tangent line to the graph of \(y = x^4 - x^3 - x^2 + x + 1\) at the point \((1, 1)\)?

The function \(f\) is continuous on the closed interval \([0, 10]\) with values \(f(0)=1\), \(f(1)=-1\), \(f(3)=4\), \(f(7)=2\), \(f(10)=3\). Using the subintervals \([0,1]\), \([1,3]\), \([3,7]\), and \([7,10]\), what is the left Riemann sum estimate for \(\displaystyle\int_{0}^{10} f(x) d x\)?

The function \(f\) is given by \(f(x) = \begin{cases} \ln 2 x & \quad \text{if } 0 < x < 2 \\ 2 \ln x & \quad \text{if } x \geq 2 \end{cases}\). The limit \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x)\) is

The differentiable function \(f(x)\) achieves its maximum when \(x = 0\). Which of the following statements must be true?
I. The function \(g(x) = x f(x)\) has a critical point when \(x = 0\).
II. The function \(h(x) = (f(x))^2\) achieves its maximum at \(x = 0\).
III. The function \(k(x) = f(x^2)\) achieves its maximum at \(x = 0\).

\(\displaystyle\int_{1}^{8} \dfrac{d x}{\sqrt[3]{x}} =\)

If \(x + y = 2\), what is the minimum value of \(x^2 + y^2\)?

A particle moves along the graph of \(y = x + \sin x\). As it passes the point \((2 \pi, 2 \pi)\), the particle's y-coordinate is increasing at a rate of \(2\) units per second. How fast is the x-coordinate of the particle changing at this point (in units per second)?

A particle moves along the x-axis. Its position at time \(t\) is given by \(x(t) = \dfrac{t^4}{24} - \dfrac{t^3}{2} + 2 t^2 - 1\). What is the maximum acceleration of the particle on the interval \(0 \leq t \leq 4\)?

The function \(f\) is given by \(f(x) = \displaystyle\int_{0}^{x} (t^2 - 3) e^t d t\). For which value of \(x\) does \(f\) have a relative minimum?

[Calculator] What is the average value of the function \(f(x) = \sin \sqrt{x}\) on the interval \([1, 3]\)?

[Calculator] Right triangle \(A B C\) has its right angle at \(A\). Leg \(A B\) is decreasing at a constant rate of \(3\) cm per minute. Leg \(A C\) is increasing at a constant rate of \(4\) cm per minute. How is the hypotenuse \(B C\) changing at the moment when \(A B = 12\) and \(A C = 5\)?

[Calculator] If \(g'(x) = x(\ln x)^2\) and \(g(2) = 3\), what is \(g(3)\)?

\(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{x^3 - 111 x^2 + 3 x - 2}{1 - 2 x + 22 x^2 + 3 x^3} =\)

[Calculator] The derivative of a function \(f\) is given by \(f'(x) = \sin(\cos x) - 0.1 x\). How many critical points does \(f\) have on the open interval \((0, 8)\)?

The function \(f\) is given by \(f(x) = \begin{cases} \dfrac{x^3}{|x|} & \quad \text{if } x \neq 0 \\ 0 & \quad \text{if } x = 0 \end{cases}\). Which of the following statements are true?
I. \(f\) is continuous at the point \(x = 0\).
II. \(f\) is differentiable at the point \(x = 0\).
III. \(x = 0\) is a point of inflection for a graph of \(f\).

[Calculator] What is the area of the region bounded by the graphs of \(y = e^{x^2}\) and \(y = \tan x\) and the vertical lines \(x = -\dfrac{1}{2}\) and \(x = 1\)?

[Calculator] What is the x-value of the point at which the tangent line to the graph of \(y = e^{x^2} + x\) is perpendicular to the line \(3 y = 2 x + 1\)?

The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the graph of \(y = 1 - x^3\). If cross-sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

The region enclosed by the graphs of \(y = e^{x - 1}\) and \(y = -x\) and the vertical lines \(x = 0\) and \(x = 2\) is rotated about the line \(y = -3\). Which of the following gives the volume of the generated solid?

[Calculator] Which of the following is an equation of the tangent line to the graph of the function \(f(x) = e^x + x^2\) at the point where \(f'(x) = 2\)?

The function \(f\) is twice differentiable on the closed interval \([0, 2]\) with \(f(0) = 1\), \(f(1) = 3\), \(f(2) = 1\). Which of the following statements is FALSE?

The function \(f\) is continuous on \([0, 3]\) with \(f(0) = 2\), \(f(1) = 5\), \(f(2) = 4\), \(f(3) = 3\). Using the subintervals \([0, 1]\), \([1, 2]\), \([2, 3]\), what is the trapezoidal approximation to \(\displaystyle\int_{0}^{3} f(x) d x\)?

[Calculator] A population of bacteria given by \(y(t)\) grows according to the equation \(\dfrac{d y}{d t} = k y\), where \(k\) is a constant and \(t\) is measured in minutes. If \(y(10) = 10\) and \(y(30) = 25\), what is the value of \(k\)?