Evaluate the limit: \(\operatorname*{lim}\limits_{x \rightarrow 5} \dfrac{\sqrt{x + 4} - 3}{x - 5}\).

Evaluate the limit: \(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\sin\left(\dfrac{\pi}{2} + h\right) - \sin\left(\dfrac{\pi}{2}\right)}{h}\).

If \(f(x) = e^{\dfrac{1}{x}}\), then \(f'(1)\) equals:

The position of a particle \(P\) on a line is given by the equation \(s(t) = t^3 + t^2 - t - 3\). On which interval is the particle moving to the right?

Define \(F(x) = \displaystyle\int_{x}^{1} \ln t d t\). Find \(F'(2)\).

Evaluate \(\int \dfrac{x}{\sqrt{9 - x^2}} d x\).

If \(f(x) = \log_2(3 x)\), then \(f'(x)\) equals

Let \(f(t) = \dfrac{1}{t^2} - 4\) and \(g(t) = \cos t\). Find the derivative of the composition \((f \circ g)(t)\).

[Calc] Given the initial value problem \(\dfrac{d y}{d x} = k y\), with conditions \(y(0) = 10\) and \(y(2) = 18\). The constant of proportionality \(k\) equals:

Let \(f\) be a twice-differentiable function (a function whose first and second derivatives both exist). \(f''(c) = 0\) could mean that

Find the slope of the tangent to the curve \(x^2 y + 3 x^2 y^3 = 4\) at the point \((1, -1)\).

Evaluate the limit \(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{4 x^2 - 8 x}{8 x^2 + 6 x + 5}\).

Use the chart below for questions 14 and 15 about the graph of a continuous function \(f\) whose first and second derivatives are also continuous. The only critical points of \(f\) are at \(x = 0\) and \(x = 2\). Chart values: at \(x = -1, 0, 1, 2, 3, 4\), \(f'\) has signs \(+, 0, +, 0, -, -\) and \(f''\) has signs \(+, 0, -, -, -, -\). The function \(f\) has a local maximum at:

Use the chart from question 14: \(f'\) signs at \(x = -1, 0, 1, 2, 3, 4\) are \(+, 0, +, 0, -, -\). The function \(f\) is decreasing on which interval(s)?

Evaluate \(\int x^3 e^x d x\).

[Calc] A rectangle is to be inscribed under one arch of the sine curve \(y = \sin x\) on \([0, \pi]\) with its base on the x-axis. What is the area of the largest rectangle that can be formed?

[Calc] Let \(L(x)\) be the linearization of the function \(f(x) = \sqrt{1 + x}\) at \(x = 0\). The difference between \(L\) and \(f\) at \(x = 0.2\) would be: