?(a) If \(f(x)=4 x-\tan x,-\pi / 2<x<\pi / 2\), find \(f^{\prime}\) and \(f^{\prime

Find points \(P\) and \(Q\) on the parabola \(y=1-x^{2}\) so that the triangle \(ABC\) formed by the \(x-axis\) and the tangent lines at \(P\) and \(Q\) is an equilateral triangle (see the figure). Equation Transcription: Text Transcription: P Q y=1-x^2 ABC x-axis

Find the point where the curves \(y=x^3-3 x+4\) and \(y=3\left(x^2-x\right)\) are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent.

Show that the tangent lines to the parabola \(y=a x^{2}+b x+c\) at any two points with \(x-coordinates\) \(p\) and \(q\) must intersect at a point whose \(x-coordinates\) is halfway between \(p\) and \(q\). Equation Transcription: Text Transcription: y=ax^2+bx+c x-coordinates p q

(a) Explain how implicit differentiation works. (b) Explain how logarithmic differentiation works.

Give several examples of how the derivative can be interpreted as a rate of change in physics, chemistry, biology, economics, or other sciences.

Find the values of the constants a and b such that \(\lim _{x \rightarrow 0} \frac{\sqrt[3]{a x+b}-2}{x}=\frac{5}{12}\)

(a) Write an expression for the linearization of f at a. (b) If y = f(x), write an expression for the differential dy. (c) If \(d x=\Delta x\), draw a picture showing the geometric meanings of \(\Delta y\) and dy.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and t are differentiable, then

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are differentiable, then \(\frac{d}{d x}[f(x) g(x)]=f^{\prime}(x) g^{\prime}(x)\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are differentiable, then \(\frac{d}{d x}[f(g(x))]=f^{\prime}(g(x)) g^{\prime}(x)\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is differentiable, then

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is differentiable, then \(\frac{d}{d x} f(\sqrt{x})=\frac{f^{\prime}(x)}{2 \sqrt{x}}\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If y = e2, then y’ = 2e.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a polynomial is a polynomial.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f(x)=\left(x^{6}-x^{4}\right)^{5}\), then \(f^{(31)}(x)=0\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a rational function is a rational function.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. An equation of the tangent line to the parabola \(y=x^{2}\) at (-2, 4) is y - 4 = 2x(x + 2).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If g(x) = x5, then

Calculate \(y^{\prime}\). \(y=\left(x^{2}+x^{3}\right)^{4}\)

Calculate y’. y =

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Calculate y’. y = x2 sin

Calculate y’. y = x cos-1 x

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Calculate y’. xey = y sin x

Calculate y’. y = ln(x ln x)

Calculate \(y^{\prime}\). \(y=e^{m x} \cos n x\) Text Transcription: y’ y = e^{mx} cos nx

Calculate y’. y =

Calculate y’. y = (arcsin 2x)2

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Calculate y’. y = ln sec x

Calculate y’. y + x cos y = x2y

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Calculate y’. y = cot (csc x)

Calculate y’. y = tan

Calculate y’. y = ex sec x

Calculate y’. y = 3x ln x

Calculate y’. y = sec(1 + x2)

Calculate y’. y = (1 - x-1)-1

Calculate \(y \prime\). \(y=1 / \sqrt[3]{x+\sqrt{x}}\)

Calculate \(y^\prime\). \(\sin (x y)=x^{2}-y\)

Calculate y’. y =

Calculate y’. y = log5(1 + 2x)

Calculate y’. y = (cos x)x

Calculate y’. y = ln sin x -

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Calculate \(y^\prime\). \(y=e^{\cos x}+\cos \left(e^{x}\right)\)

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Calculate \(y^\prime\). \(y=\sqrt{t \ln \left(t^{4}\right)}\)

Calculate \(y^\prime\). \(y=\sin \left(\tan \sqrt{1+x^{3}}\right)\)

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Calculate \(y \prime\). \(y=\tan ^2(\sin \theta)\)

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Calculate \(y \prime\). \(y=x \sinh \left(x^2\right)\)

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If , find

Find if

Find \(f^{(n)}(x)\) if f(x)=1/(2-x)

Use mathematical induction to show that if , then (Note: See Principles of Problem Solving following Chapter 1.)

Evaluate .

Find an equation of the tangent line to the curve at the given point.

Find an equation of the tangent line to the curve at the given point.

Find an equation of the tangent line to the curve at the given point.

Find equations of the tangent line and normal line to the curve at the given point. \(x^{2}+4 x y+y^{2}=13\), (2, 1)

Find equations of the tangent line and normal line to the curve at the given point.

If f(x) = xesin x, find f’(x). Graph f and f’ on the same screen and comment.

(a) If \(f(x)=x \sqrt{5-x}\), find \(f^{\prime}(x)\). (b) Find equations of the tangent lines to the curve \(y=x \sqrt{5-x}\), at the points (1, 2) and (4, 4). (c) Illustrate part (b) by graphing the curve and tangent lines on the same screen. (d) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and \(f^\prime\).

(a) If \(f(x)=4 x-\tan x,-\pi / 2<x<\pi / 2\), find \(f^{\prime}\) and \(f^{\prime \prime}\). (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f, \(f^{\prime}\), and \(f^{\prime \prime}\).

At what points on the curve y = sin x + cos x, 0 ? x ? 2, is the tangent line horizontal?

Find the points on the ellipse x2 + 2y2 = 1 where the tangent line has slope 1.

If f(x) = (x - a)(x - b)(x - c), show that

(a) By differentiating the double-angle formula cos 2x = cos2x - sin2x obtain the double-angle formula for the sine function. (b) By differentiating the addition formula sin(x + a) = sin x cos a + cos x sin a obtain the addition formula for the cosine function.

Suppose that f(1) = 2 f’(1) = 3 f(2) = 1 f’(2) = 2 g(1) = 3 g’(1) = 1 g(2) = 1 g’(2) = 4 (a) If S(x) = f(x) + g(x), find S’(1). (b) If P(x) = f(x)g(x), find P’(2). (c) If Q(x) = f(x)/g(x), find Q’(1). (d) If C(x)= f(t(x)), find C’(2).

If f and g are the functions whose graphs are shown, let P(x) = f(x) g(x), Q(x) = f(x) / g(x), and C(x)=f(g(x)). Find (a) \(P^{\prime}(2)\), (b) \(Q^{\prime}(2)\), and (c) \(C^{\prime}(2)\).

Find f’ in terms of g’. f(x) = x2g(x)

Find \(f^\prime\) in terms of \(g^\prime\). \(f(x)=g\left(x^{2}\right)\)

Find f’ in terms of g’. f(x) = [g(x)]2

Find f’ in terms of g’. f(x) = g(g(x))

Find \(f \prime\) in terms of \(g \prime\). \(f(x) = g(e^x)\)

Find \(f^\prime\) in terms of \(g^\prime\). \(f(x)=e^{g(x)}\)

Find f’ in terms of g’. f(x) = ln

Find f’ in terms of g’. f(x = g(ln x)

Find h’ in terms of f’ and g’. h(x) =

Find h’ in terms of f’ and g’. h(x) =

Find h’ in terms of f’ and g’. h(x) = f(g(sin 4x))

(a) Graph the function f(x) = x - 2 sin x in the viewing rectangle [0, 8] by [-2, 8]. (b) On which interval is the average rate of change larger: [1, 2] or [2, 3]? (c) At which value of x is the instantaneous rate of change larger: x = 2 or x = 5? (d)Check your visual estimates in part (c) by computing f’(x) and comparing the numerical values of f’(2) and f’(5).

At what point on the curve \(y=[\ln (x+4)]^{2}\) is the tangent horizontal? Equation Transcription: Text Transcription: y=[ln(x+4)]^2

(a) Find an equation of the tangent line to the curve \(y=e^{x}\) that is parallel to the line \(x-4 y=1\). (b) Find an equation of the tangent to the curve \(y=e^{x}\) that passes through the origin. Equation Transcription: Text Transcription: y=e^x x - 4y = 1

Find a parabola \(y=a x^2+b x+c\) that passes through the point (1,4) and whose tangent lines at x=-1 and x=5 have slopes 6 and -2, respectively.

The function \(C(t)=K\left(e^{-a t}-e^{-b t}\right)\), where \(a,b\), and \(K\) are positive constants and \(b>a\), is used to model the concentration at time \(t\) of a drug injected into the bloodstream. (a) Show that \(\lim _{t \rightarrow \infty} C(t)=0\). (b) Find \(C^{\prime}(t)\), the rate of change of drug concentration in the blood. (c) When is this rate equal to \(0\)? Equation Transcription: C’(t) 0 Text Transcription: C(t)=K(e^-at - e^-bt) a b K b>a t lim_t right arrow infinity C(t)=0 C^prime(t) 0

An equation of motion of the form \(s=A e^{-c t} \cos (\omega t+\delta)\) represents damped oscillation of an object. Find the velocity and acceleration of the object. Equation Transcription: ???? text transcription: s=Ae^-ct cos(omega t + delta)

A particle moves along a horizontal line so that its coordinate at time \(t\) is \(x=\sqrt{b^{2}+c^{2} t^{2}}, t \geqslant 0\), where \(b\) and \(c\) are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive direction. Equation Transcription: Text Transcription: t x=b^2 + c^2 t^2, t geqslant 0 b c

A particle moves on a vertical line so that its coordinate at time \(t\) is \(y=t^{3}-12 t+3, t \geqslant 0\). (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval \(0 \leqslant t \leqslant 3\). (d) Graph the position, velocity, and acceleration functions for \(0 \leqslant t \leqslant 3\). (e) When is the particle speeding up? When is it slowing down? Equation Transcription: Text Transcription: t y=t^3 - 12t + 3, t geqslant 0 0 leqslant t leqslant 3

The volume of a right circular cone is \(V=\frac{1}{3} \pi r^{2} h\), where \(r\) is the radius of the base and \(h\) is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant. Equation Transcription: Text Transcription: V=1/3 pi r^2 h r h

The mass of part of a wire is \(x(1+\sqrt{x})\) kilograms, where \(x\) is measured in meters from one end of the wire. Find the linear density of the wire when \(x=4 m\). Equation Transcription: Text Transcription: x(1+ sqrt x) x x=4 m

The cost, in dollars, of producing \(x\) units of a certain commodity is \(C(x)=920+2 x-0.02 x^{2}+0.00007 x^{3}\) (a) Find the marginal cost function. (b) Find \(C^{\prime}(100)\) and explain its meaning. (c) Compare \(C^{\prime}(100)\) with the cost of producing the \(101st\) item. Equation Transcription: Text Transcription: x C(x)=920+2x-0.02x^2+0.00007x^3 C^prime(100) 101st

A bacteria culture contains \(200\) cells initially and grows at a rate proportional to its size. After half an hour the population has increased to \(360\) cells. (a) Find the number of cells after \(t\) hours. (b) Find the number of cells after \(4\) hours. (c) Find the rate of growth after \(4\) hours. (d) When will the population reach \(10,000\)? Equation Transcription: Text Transcription: 200 360 t 4 10,000

Cobalt-\(60\) has a half-life of \(5.24\) years. (a) Find the mass that remains from a \(100-mg\) sample after \(20\) years. (b) How long would it take for the mass to decay to \(1 mg\)? Equation Transcription: Text Transcription: 60 5.24 100-mg 20 1 mg

Let \(C(t)\) be the concentration of a drug in the bloodstream. As the body eliminates the drug, \(C(t)\) decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus \(C^{\prime}(t)=-k C(t)\), where \(k\) is a positive number called the elimination constant of the drug. (a) If \(C_{0}\) is the concentration at time \(t=0\), find the concentration at time \(t\). (b) If the body eliminates half the drug in \(30\) hours, how long does it take to eliminate \(90 \%\) of the drug? Equation Transcription: Text Transcription: C(t) C^prime (t)=-kC(t) k C_0 t=0 t 30 90%

A cup of hot chocolate has temperature \(80^{\circ} \mathrm{C}\) in a room kept at \(20^{\circ} \mathrm{C}\). After half an hour the hot chocolate cools to \(60^{\circ} \mathrm{C}\). (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to \(40^{\circ} \mathrm{C}\) ? Equation Transcription: Text Transcription: 80^circ C 20^circ C 60^circ C 40^circ C

The volume of a cube is increasing at a rate of \(10 \mathrm{~cm}^{3} / \mathrm{min}\). How fast is the surface area increasing when the length of an edge is \(30 \mathrm{~cm}\) ? Equation Transcription: Text Transcription: 10 cm^3 / min 30 cm

A paper cup has the shape of a cone with height \(10 cm\) and radius \(3 cm\) (at the top). If water is poured into the cup at a rate of \(2 \mathrm{~cm}^{3} / \mathrm{s}\), how fast is the water level rising when the water is \(5 cm\) deep? Equation Transcription: Text Transcription: 10 cm 3 cm 2 cm^3 /s 5 cm

A balloon is rising at a constant speed of \(5 \mathrm{ft} / \mathrm{s}\). A boy is cycling along a straight road at a speed of \(15 \mathrm{ft} / \mathrm{s}\). When he passes under the balloon, it is \(45 ft\) above him. How fast is the distance between the boy and the balloon increasing \(3 s\) later? Equation Transcription: Text Transcription: 5 ft/s 15 ft/s 45 ft 3 s

A waterskier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the ramp?

The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is \(\pi / 6\)?

(a) Find the linear approximation to \(f(x)=\sqrt{25-x^{2}}\) near 3. (b) Illustrate part (a) by graphing f and the linear approximation. (c) For what values of x is the linear approximation accurate to within 0.1?

(a) Find the linearization of \(f(x)=\sqrt[3]{1+3 x}\) at a=0. State the corresponding linear approximation and use it to give an approximate value for \(\sqrt[3]{1.03}\). (b) Determine the values of x for which the linear approximation given in part (a) is accurate to within 0.1.

Evaluate dy if \(y=x^{3}-2 x^{2}+1\), x=2, and dx=0.2.

A window has the shape of a square surmounted by a semi- circle. The base of the window is measured as having width \(60 cm\) with a possible error in measurement of \(0.1 cm\). Use differentials to estimate the maximum error possible in computing the area of the window. Equation Transcription: Text Transcription: 60 cm 0.1 cm

Express the limit as a derivative and evaluate. \(\lim \limits_{x \rightarrow 1} \frac{x^{17}-1}{x-1}\) Equation Transcription: Text Transcription: lim_x right arrow 1 x^17-1/x - 1

Express the limit as a derivative and evaluate. \(\lim \limits_{h \rightarrow 0} \frac{\sqrt[4]{16+h}-2}{h}\) Equation Transcription: Text Transcription: lim_h right arrow 0 4th root 16 + h - 2/h

Express the limit as a derivative and evaluate. \(\lim \limits_{\theta \rightarrow \pi / 3} \frac{\cos \theta-0.5}{\theta-\pi / 3}\) Equation Transcription: Text Transcription: lim_theta right arrow pi/3 cos theta - 0.5 - theta - pi/3

Evaluate \(\lim \limits_{x \rightarrow 0} \frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^{3}}\). Equation Transcription: Text Transcription: lim_x right arrow 0 sqrt 1+tan x - sqrt 1+sin x / x^3

Suppose \(f\) is a differentiable function such that \(f(g(x))=x\) and \(f^{\prime}(x)=1+[f(x)]^{2}\). Show that \(g^{\prime}(x)=1 /\left(1+x^{2}\right)\). Equation Transcription: Text Transcription: f f(g(x))=x f^prime(x)=1+[f(x)]^2 g^prime(x)=1/(1+x^2)

Find \(f^{\prime}(x)\) if it is known that \(\frac{d}{d x}[f(2 x)]=x^{2}\) Equation Transcription: Text Transcription: f^prime(x) d/dx [f(2x)]=x^2

Show that the length of the portion of any tangent line to the astroid \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\) cut off by the coordinate axes is constant. Equation Transcription: Text Transcription: x^2/3 + y^2/3 = a^2/3

Find points and on the parabola so that the triangle formed by the -axis and the tangent lines at and is an equilateral triangle (see the figure).

Find the point where the curves and are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent.

Show that the tangent lines to the parabola at any two points with -coordinates and must intersect at a point whose -coordinate is halfway between and .

Show that \(\frac{d}{d x}\left(\frac{\sin ^2 x}{1+\cot x}+\frac{\cos ^2 x}{1+\tan x}\right)=-\cos 2 x\).

If \(f(x)=\lim \limits_{t\rightarrow x}\frac{\sec t-\sec x}{t-x}\) , find the value of \(f^{\prime}(\pi / 4)\). Equation Transcription: Text Transcription: f(x)=lim_t rightarrow x sec t - sec x / t-x f^prime(pi/4)

Find the values of the constants a and b such that \(\lim _{x \rightarrow 0} \frac{\sqrt[3]{a x+b}-2}{x}=\frac{5}{12}\)

Show that \(\sin ^{-1}(\tanh x)=\tan ^{-1}(\sinh x)\).

A car is traveling at night along a highway shaped like a parabola with its vertex at the origin (see the figure). The car starts at a point \(100 m\) west and \(100 m\) north of the origin and travels in an easterly direction. There is a statue located \(100 m\) east and \(50 m\) north of the origin. At what point on the highway will the car's headlights illuminate the statue? Equation Transcription: Text Transcription: 100 m 50 m

Prove that \(\frac{d^{n}}{d x^{n}}\left(\sin ^{4} x+\cos ^{4} x\right)=4^{n-1} \cos (4 x+n \pi / 2)\) Equation Transcription: Text Transcription: d^n/dx^n (sin^4 x + cos^4 x)=4^n-1 cos(4x+npi/2)

If \(f\) is differentiable at \(a\), where \(a>0\), evaluate the following limit in terms of \(f^{\prime}(a)\) : \(\lim \limits_{x \rightarrow a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}}\) Equation Transcription Text Transcription: f a a>0 f^prime (a) lim_x right arrow a f(x)-f(a)/ sqrt x - sqrt a

The figure shows a circle with radius 1 inscribed in the parabola \(y=x^{2}\). Find the center of the circle.

Find all values of \(c\) such that the parabolas \(y=4 x^{2}\) and \(x=c+2 y^{2}\) intersect each other at right angles. Equation Transcription: Text Transcription: c y=4x^2 x=c+2y^2

How many lines are tangent to both of the circles \(x^{2}+y^{2}=4\) and \(x^{2}+(y-3)^{2}=1\)? At what points do these tangent lines touch the circles? Equation Transcription: Text Transcription: x^2+y^2=4 x^2+(y-3)^2=1

If \(f(x)=\frac{x^{46}+x^{45}+2}{1+x}\), calculate \(f^{(46)}(3)\). Express your answer using factorial notation: \(n !=1 \cdot 2 \cdot 3 \cdots \cdots(n-1) \cdot n\) Equation Transcription: Text Transcription: f(x)=x^46+ x^45 + 2 /1 + x f^(46)(3) n!=1 cdot 2 cdot 3 cdots (n-1) cdot n

The figure shows a rotating wheel with radius \(40 cm\) and a connecting rod \(AP\) with length \(1.2 m\). The pin \(P\) slides back and forth along the \(x-axis\) as the wheel rotates counterclockwise at a rate of \(360\) revolutions per minute. (a) Find the angular velocity of the connecting rod, \(d \alpha / d t\), in radians per second, when \(\theta=\pi / 3\). (b) Express the distance \(x=|O P|\) in terms of \(\theta\) (c) Find an expression for the velocity of the pin \(P\) in terms of \(\theta\). Equation Transcription: Text Transcription: 40 cm AP 1.2 m P x-axis 360 d alpha /dt theta = pi/3 x=|OP| theta

Tangent lines \(T_{1}\) and \(T_{2}\) are drawn at two points \(P_{1}\) and \(P_{2}\) on the parabola \(y=x^{2}\) and they intersect at a point \(P\). Another tangent line \(T\) is drawn at a point between \(P_{1}\) and \(P_{2}\); it intersects \(T_{1}\) at \(Q_{1}\) and \(T_{2}\) at \(Q_{2}\). Show that \(\frac{\left|P Q_{1}\right|}{\left|P P_{1}\right|}+\frac{\left|P Q_{2}\right|}{\left|P P_{2}\right|}=1\) Equation Transcription: Text Transcription: T_1 T_2 P_1 P_2 y=x^2 P T Q_1 Q_2 |PQ1|/|PP1| + |PQ2|/|PP2| = 1

Show that \(\frac{d}{d r}\left(e^{a x} \sin b x\right)=r^{n} e^{a x} \sin (b x+n \theta)\) where \(a\) and \(b\) are positive numbers, \(r^{2}=a^{2}+b^{2}\), and \(\theta=\tan ^{-1}(b / a)\). Equation Transcription: Text Transcription: frac d^n (e^ax sin bx)=r^n e^ax sin (bx+n theta) a b r^2=a^2+b^2 theta=tan^-1 (b/a)

Evaluate \(\lim \limits_{x \rightarrow \pi} \frac{e^{\sin x}-1}{x-\pi}\). Equation Transcription: Text Transcription: lim_x right arrow pi e^sinx -1/x-pi

Let \(T\) and \(N\) be the tangent and normal lines to the ellipse \(x^{2} / 9+y^{2} / 4=1\) at any point \(P\) on the ellipse in the first quadrant. Let \(x_{T}\) and \(y_{T}\) be the \(x\)-and \(y\)-intercepts of \(T\) and \(x_{N}\) and \(y_{N}\) be the intercepts of \(N\). As \(P\) moves along the ellipse in the first quadrant (but not on the axes), what values can \(x_{T}, y_{T}, x_{N}\), and \(y_{N}\) take on? First try to guess the answers just by looking at the figure. Then use calculus to solve the problem and see how good your intuition is. Equation Transcription: Text Transcription: T N x^2 /9 + y^2 /4=1 P x_T y_T x y x_N y_N N

Evaluate \(\lim \limits_{x \rightarrow 0} \frac{\sin (3+x)^{2}-\sin 9}{x}\) . Equation Transcription: Text Transcription: lim_x right arrow 0 sin(3+x)^2-sin 9/x

(a) Use the identity for tan(x - y) (see Equation 15b in Appendix D) to show that if two lines \(L_{1}\) and \(L_{2}\) intersect at an angle \(\alpha\), then \(\tan \alpha=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}}\) where \(m_{1} \text { and } m_{2}\) are the slopes of \(L_{1}\) and \(L_{2}\), respectively. (b) The angle between the curves \(C_{1}\) and \(C_{2}\) at a point of intersection P is defined to be the angle between the tangent lines to \(C_{1}\) and \(C_{2}\) at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection. (i) \(y = x^2\) and \(y = (x - 2)^2\) (ii) \(x^{2}-y^{2}=3\) and \(x^{2}-4 x+y^{2}+3=0\)

Let \(P\left(x_{1}, y_{1}\right)\) be a point on the parabola \(y^{2}=4 p x\) with focus \(F(p, 0)\). Let \(\alpha\) be the angle between the parabola and the line segment \(FP\), and let \(\beta\) be the angle between the horizontal line \(y=y_{1}\) and the parabola as in the figure. Prove that \(\alpha=\beta\). (Thus, by a principle of geometrical optics, light from a source placed at \(F\) will be reflected along a line parallel to the \(x-axis\). This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.) Equation Transcription: Text Transcription: P(x_1,y_1) y^2=4px F(p,0) alpha FP beta y=y_1 alpha=beta F x-axis

Suppose that we replace the parabolic mirror of Problem 22 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle with center O. A ray of light coming in toward the mirror parallel to the axis along the line PQ will be reflected to the point R on the axis so that \(\angle P Q O=\angle O Q R\) (the angle of incidence is equal to the angle of reflection). What happens to the point R as P is taken closer and closer to the axis?

If \(f\) and \(g\) are differentiable functions with \(f(0)=g(0)=0\) and \(g^{\prime}(0) \neq 0\) show that \(\lim \limits_{x \rightarrow 0} \frac{f(x)}{g(x)}=\frac{f^{\prime}(0)}{g^{\prime}(0)}\) Equation Transcription: Text Transcription: f g f(0)=g(0)=0 g^prime (0) neq 0 lim_x right arrow 0 f(x)/g(x) = f^prime(0)/g^prime(0)

Evaluate \(\lim _{x \rightarrow 0} \frac{\sin (a+2 x)-2 \sin (a+x)+\sin a}{x^2}\).

(a) The cubic function \(f(x)=x(x-2)(x-6)\) has three distinct zeros: \(0,2\), and \(6\). Graph \(f\) and its tangent lines at the average of each pair of zeros. What do you notice? (b) Suppose the cubic function \(f(x)=(x-a)(x-b)(x-c)\) has three distinct zeros: \(a,b\), and \(c\). Prove, with the help of a computer algebra system, that a tangent line drawn at the average of the zeros \(a\) and \(b\) intersects the graph of \(f\) at the third zero. Equation Transcription: Text Transcription: f(x)=x(x-2)(x-6) 0,2, and 6 f f(x)=(x-a)(x-b)(x-c) a,b, and c a and b

For what value of \(k\) does the equation \(e^{2 x}=k \sqrt{x}\) have exactly one solution? Equation Transcription: Text Transcription: k e^2x=k sqrt x

For which positive numbers a is it true that \(a^{x} \geqslant 1+x\) for all x?

If \(y=\frac{x}{\sqrt{a^{2}-1}}-\frac{2}{\sqrt{a^{2}-1}} \arctan \frac{\sin x}{a+\sqrt{a^{2}-1}+\cos x}\) show that \(y^{\prime}=\frac{1}{a+\cos x}\) Equation Transcription: Text Transcription: y=x/sqrt a^2-1 - 2/sqrt a^2-1 arctan sin x/a+sqrt a^2-1 + cos x y^prime= 1/a+cos x

Given an ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\), where \(a \neq b\), find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals.

Find the two points on the curve \(y=x^{4}-2 x^{2}-x\) that have a common tangent line. Equation Transcription: Text Transcription: y=x^4-2x^2-x

Suppose that three points on the parabola \(y=x^{2}\) have the property that their normal lines intersect at a common point. Show that the sum of their \(x-coordinates\) is \(0\). Equation Transcription: Text Transcription: y=x^2 x-coordinates 0

A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with slope \(\frac{2}{5}\) intersects some of these circles.

A cone of radius \(r\) centimeters and height \(h\) centimeters is lowered point first at a rate of \(1 \mathrm{~cm} / \mathrm{s}\) into a tall cylinder of radius \(R\) centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged? Equation Transcription: Text Transcription: r h 1 cm/s R

A container in the shape of an inverted cone has height \(16 cm\) and radius \(5 cm\) at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surface area of a cone is \(\pi r l\), where \(r\) is the radius and \(l\) is the slant height.) If we pour the liquid into the container at a rate of \(2 \mathrm{~cm}^{3} / \mathrm{min}\), then the height of the liquid decreases at a rate of \(0.3 \mathrm{~cm} / \mathrm{min}\) when the height is \(10 cm\). If our goal is to keep the liquid at a constant height of \(10 cm\), at what rate should we pour the liquid into the container? Equation Transcription: Text Transcription: 16 cm 5 cm rl r l 2 cm^3/min 0.3 cm/min 10 cm