Problem 43
Explorations Hyperbolas Let \(x=a \sec t\) and \(y=b \tan t\) (a) Writing to Learn Let \(a=1,2,\) or \(3, b=1,2,\) or \(3,\) and graph using the parameter interval \((-\pi / 2, \pi / 2)\) . Explain what you see, and describe the role of \(a\) and \(b\) in these parametric equations. (Caution: If you get what appear to be asymptomes, try using the approximation \([-1.57,1.57]\) for the parameter interval.) (b) Let \(a=2, b=3,\) and graph in the parameter interval \((\pi / 2,3 \pi / 2)\) . Explain what you see. (c) Writing to Learn Let \(a=2, b=3,\) and graph using the parameter interval \((-\pi / 2,3 \pi / 2) .\) Explain why you must be careful about graphing in this interval or any interval that contains \(\pm \pi / 2\) . (d) Use algebra to explain why \(\left(\frac{x}{a}\right)^{2}-\left(\frac{y}{b}\right)^{2}=1\) (e) Let \(x=a\) tan \(t\) and \(y=b\) sec \(t .\) Repeat (a), (b), and (d) using an appropriate version of \((\mathrm{d}) .\)
Problem 43
Pressure under Water The pressure \(p\) experienced by a diver under water is related to the diver's depth \(d\) by an equation of the form \(p=k d+1(k\) a constant). When \(d=0\) meters, the pressure is 1 atmosphere. The pressure at 100 meters is 10.94 atmospheres. Find the pressure at 50 meters.
Problem 44
Modeling Distance Traveled A car starts from point \(P\) at time \(t=0\) and travels at 45 mph. (a)Write an expression \(d(t)\) for the distance the car travels from \(P\) (b) Graph \(y=d(t) .\) (c) What is the slope of the graph in (b)? What does it have to do with the car? (d) Writing to Learn Create a scenario in which \(t\) could have negative values. (e) Writing to Learn Create a scenario in which the \(y\) -intercept of \(y=d(t)\) could be \(30 .\)
Problem 44
Multiple Choice Which of the following gives the domain of \(y=2 e^{-x}-3 ?\) \((\mathbf{A})(-\infty, \infty) \quad(\mathbf{B})[-3, \infty) \quad(\mathbf{C})[-1, \infty) \quad(\mathbf{D})(-\infty, 3]\) \((\mathbf{E}) x \neq 0\)
Problem 44
Transformations Let \(x=(2 \cos t)+h\) and \(y=(2 \sin t)+k\) (a) Writing to Learn Let \(k=0\) and \(h=-2,-1,1,\) and \(2,\) in turn. Graph using the parameter interval \([0,2 \pi]\) . Describe the role of \(h\) (b) Writing to Learn Let \(h=0\) and \(k=-2,-1,1,\) and \(2,\) in turn. Graph using the parameter interval \([0,2 \pi] .\) Describe the role of \(k\) (c) Find a parametrization for the circle with radius 5 and center at \((2,-3)\)(d) Find a parametrization for the ellipse centered at \((-3,4)\) with semi major axis of length 5 parallel to the \(x\) -axis and semi-minor axis of length 2 parallel to the \(y\) -axis (d) Find a parametrization for the ellipse centered at \((-3,4)\) with semi major axis of length 5 parallel to the \(x\) -axis and semi minor axis of length 2 parallel to the \(y\) -axis.
Problem 44
In Exercises 43 and \(44,\) find a formula for \(f^{-1}\) and verify that \(\left(f \circ f^{-1}\right)(x)=\left(f^{-1} \circ f\right)(x)=x\). $$f(x)=\frac{50}{1+1.1^{-x}}$$
Problem 45
Multiple Choice Which of the following gives the range of \(y=4-2^{-x} ?\) \((\mathbf{A})(-\infty, \infty) \quad(\mathbf{B})(-\infty, 4) \quad(\mathbf{C})[-4, \infty)\) \((\mathbf{D})(-\infty, 4]\) (E) all reals
Problem 45
In Exercises 45 and \(46,\) a parametrization is given for a curve.
(a) Graph the curve. What are the initial and terminal points, if any?
Indicate the direction in which the curve is traced.
(b) Find a Cartesian equation for a curve that contains the parametrized
curve. What portion of the graph of the Cartesian equation is traced by the
parametrized curve?
$$x=-\sec t, \quad y=\tan t, \quad-\pi / 2
Problem 45
Even-Odd (a) Show that cot \(x\) is an odd function of \(x\) . (b) Show that the quotient of an even function and an odd function is an odd function.
Problem 45
Self-inverse Prove that the function \(f\) is its own inverse. (a) \(f(x)=\sqrt{1-x^{2}}, \quad x \geq 0 \quad\) (b) \(f(x)=1 / x\)
