Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). \(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{(2 + h)^5 - 32}{h}\), \(h = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001\)

Consider the function \(f(x) = \tan\left(\dfrac{1}{x}\right)\).

Explain in your own words what is meant by the equation \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x) = 5\). Is it possible for this statement to be true and yet \(f(2) = 3\)? Explain.

Use the Intermediate Value Theorem to show that there is a solution of the equation in the given interval. \(\dfrac{2}{x} = x - \sqrt{x}\), \((2, 3)\)

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(B(u) = \sqrt{3u - 2} + \sqrt{2u - 3}\)

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} \cos x & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ 1 - x^2 & \text{if } x > 0 \end{cases}\), \(a = 0\)

If \(f(x) = x^2 + 10 \sin x\), show that there is a number \(c\) such that \(f(c) = 1000\).

Suppose \(f\) and \(g\) are continuous functions such that \(g(2) = 6\) and \(\operatorname*{lim}\limits_{x \rightarrow 2} [3 f(x) + f(x) g(x)] = 36\). Find \(f(2)\).

Find the given derivative by finding the first few derivatives and observing the pattern that occurs. \(D^{103} \cos 2x\)

Write the composite function in the form \(f(g(x))\). [Identify the inner function \(u = g(x)\) and the outer function \(y = f(u)\).] Then find the derivative \(\dfrac{d y}{d x}\). \(y = \tan(x^2)\)

Find the derivative of the function. \(A(t) = \dfrac{1}{(\cos t + \tan t)^2}\)

Find the derivative of the function. \(G(z) = (1 - 4z)^2 \sqrt{z^2 + 1}\)

The function \(f(x) = \sin(x + \sin 2x)\), \(0 \leq x \leq \pi\), arises in applications to frequency modulation (FM) synthesis.

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(y \sin 2x = x \cos 2y\), \(\quad \left(\dfrac{\pi}{2}, \dfrac{\pi}{4}\right)\)

If \(f(x) + x^2 [f(x)]^3 = 10\) and \(f(1) = 2\), find \(f'(1)\).

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. \(x^2 + y^2 = r^2\), \(\quad a x + b y = 0\)

Use implicit differentiation to find \(\dfrac{d y}{d x}\) for the equation \(\dfrac{x}{y} = y^2 + 1\) (where \(y \neq 0\)). Then show that you get the same answer when you perform implicit differentiation on the equivalent equation \(x = y^3 + y\) (where \(y \neq 0\)).

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(y^2(6 - x) = x^3\), \(\quad (2, \sqrt{2})\) (cissoid of Diocles)