Chapter 0 Challenge Problems

Let \(g(x) = x^2 + bx + 2\). For what values of \(b\) does \(g\) have no real zeros?

A \(100\)-point exam has \(16\) questions, each worth either four or seven points. Determine how many four-point questions and seven-point questions are on the exam.

\(\hspace{0.56em}3.\)

For any positive integer \(n,\) let \(f(x) = x^n.\) Prove that

\(f\) is even if \(n\) is even
\(f\) is odd if \(n\) is odd

\(\hspace{0.56em}4.\)

Find the value of \(k\) such that the points \((-3, 4),\) \((1, 1),\) and \((k, -2)\) are collinear.

\(\hspace{0.56em}5.\)

A rectangular prism has a square base whose lengths are \(x.\) If the prism's volume is \(400,\) then express its surface area \(S\) as a function of \(x.\) Then give the domain in context.

\(\hspace{0.56em}6.\)

Solve for \(x\) in the equation \[\frac{3}{\csc x} - \frac{6}{\cot x} = 0 \pd\]

\(\hspace{0.56em}7.\)

Particle A moves to the right with a speed of \(4\) feet per second. Particle B, initially located \(20\) feet to the right of particle A, travels to the left with a speed of \(6\) feet per second. When do both particles collide?

\(\hspace{0.56em}8.\)

In Figure 1 find the side lengths \(d_1,\) \(d_2,\) \(d_3,\) and \(d_4\) and the angles \(\theta_1,\) \(\theta_2,\) and \(\theta_3.\)

\(\hspace{0.56em}9.\)

Solve for \(x\) in \(\ds \log_7 x + \log_7(x - 4) = 1.\)

\(10.\)

Let \[f(x) = \tfrac{1}{2} x - 1 \and g(x) = 4^x \pd\]

Find \((f \circ g)(x)\) and \((g \circ f)(x)\) and their domains.
For what \(x\) is \((f \circ g)(x)\) positive?
Solve the inequality \((g \circ f)(x) \gt (f \circ g)(x) + 1.\)

\(11.\)

Show that \[\sin(\acos x) = \sqrt{1 - x^2} \pd\]

\(12.\)

A belt is wrapped around the top half of a circular shaft, and the belt is pulled at the left end with a force \(T_1.\) A frictional force opposes the shaft's rotation, so the right end of the belt experiences a tensile force \(T_2.\) (See Figure 2.) The coefficient of friction between the belt and shaft is \(\mu;\) large values of \(\mu\) indicate high levels of friction. Then \(T_1,\) \(T_2,\) and \(\mu\) satisfy \[\ln \par{\frac{T_1}{T_2}} = \mu \pi \pd\] Solve for \(T_1\) in terms of \(T_2\) and \(\mu.\) Then calculate \(T_2\) if \(T_1 = 60\) pounds and \(\mu = 0.6.\)

\(13.\)

A slot is constructed by connecting two semicircles to two horizontal line segments (Figure 3). If the slot's perimeter is \(50,\) then express its area \(A\) as a function of each semicircle's diameter, \(x.\)

\(14.\)

Using a Proof by Induction, prove the formula \[\sum_{i = 1}^n i^3 = \parbr{\frac{n(n + 1)}{2}}^2 \pd\]

\(15.\)

Some data trends are best modeled by an exponential function \(y = ab^x,\) such as the model in Figure 4A. The logarithm of the \(y\)-values is plotted against the original \(x\)-values to produce a linear pattern; the equation of the best-fit line is \[\log y = -0.6482 + 0.3633 x \pd\] (See Figure 4B.) Find the values of \(a\) and \(b\) to complete the exponential model.

\(16.\)

For two angles \(\alpha\) and \(\beta,\) it is known that \[ \ba 5 \sin \alpha + 2 \cos \beta &= 6 \cma \nl 5 \cos \alpha + 2 \sin \beta &= 1 \pd \ea \] If \(0 \lt \alpha + \beta \lt \pi/2,\) then find \(\sin(\alpha + \beta)\) and \(\cos(\alpha + \beta).\)