- (a) Plot the curve r = θ/2π−θ for 0 ≤ θ(b) With r as in (a), compute the limits (c) Explain how the limits in (b) show that the curve approaches a horizontal asymptote as θ approaches 2π
- If we rewrite the general equation of degree 2 (Eq. 12) in terms of variables x̃ and ỹ that are related to x and y by Eqs. (13) and (14), we obtain a new equation of degree 2 in x̃ and ỹ of the
- Use the geometric description in Theorem 1 to prove Theorem 2 (iii): v × w = 0 if and only if w = λv for some scalar λ or v = 0. THEOREM 1 Geometric Description of the Cross Product Given two
- Formulate and prove analogs of the result in Exercise 59 for the i- and j-components of v × w.Data From Exercise 59The components of the cross product have a geometric interpretation. Show that the
- Prove that v × w = v × u if and only if u = w + λv for some scalar λ. Assume that v ≠ 0.
- Use Exercise 77 to prove the identityData From Exercise 77Let a, b, c be nonzero vectors. Assume that b and c are not parallel, and set (axb)xc-ax(bxc) = (a - b)c - (b.c)a
- Which quadric surfaces have both hyperbolas and parabolas as traces?
- The torque about the origin O due to a force F acting on an object with position vector r is the vector quantity τ = r × F. If several forces Fj act at positions rj, then the net torque (units: N-m
- Use Eq. (2) to evaluate ∫ cosh4 x dx. +"=1 / cosh"-2 x dx 71 [cosh" xdx==cosh" 1x sinh x +
- Use Exercise 23 to find a general formula for the length of a helix of radius R and height h that makes N complete turns.Data From Exercise 23Let(a) Show that r(t) parametrizes a helix of radius R
- Use Green's Theorem to evaluate the line integral around the given closed curve.\(\oint_{C} x y^{3} d x+x^{3} y d y\), where \(C\) is the boundary of the rectangle \(-1 \leq x \leq 2,-2 \leq y \leq
- Use Green's Theorem to evaluate the line integral around the given closed curve.∮C(3x+5y−cosy)dx+xsinydy, where C is any closed curve enclosing a region with area 4 , oriented counterclockwise
- Use Green's Theorem to evaluate the line integral around the given closed curve.\(\oint_{C} y^{2} d x-x^{2} d y\), where \(C\) consists of the arcs \(y=x^{2}\) and \(y=\sqrt{x}, 0 \leq x \leq 1\),
- Use Green's Theorem to evaluate the line integral around the given closed curve.\(\oint_{C} y e^{x} d x+x e^{y} d y\), where \(C\) is the triangle with vertices \((-1,0),(0,4)\), and \((0,1)\),
- Let \(\mathbf{r}(t)=\left\langle t^{2}(1-t), t(t-1)^{2}ightangle\).(a) GU Plot the path \(\mathbf{r}(t)\) for \(0 \leq t \leq 1\).(b) Calculate the area \(A\) of the region enclosed by
- Calculate the area of the region bounded by the two curves \(y=x^{2}\) and \(y=4\) using the formula \(A=\oint_{C} x d y\).
- Calculate the area of the region bounded by the two curves \(y=x^{2}\) and \(y=\sqrt{x}\) for \(x \geq 0\) using the formula \(A=\oint_{C} x d y\).
- Calculate the area of the region bounded by the two curves \(y=x^{2}\) and \(y=4\) using the formula \(A=\oint_{C}-y d x\).
- Calculate the area of the region bounded by the two curves \(y=x^{2}\) and \(y=\sqrt{x}\) for \(x \geq 0\) using the formula \(A=\oint_{C}-y d x\).
- In (a)-(d), state whether the equation is an identity (valid for all \(\mathbf{F}\) or \(f\) ). If it is not, provide an example in which the equation does not hold.(a) \(\operatorname{curl}(abla
- Let \(\mathbf{F}(x, y)=\left\langle x^{2} y, x y^{2}ightangle\) be the velocity vector field for a fluid in the plane. Find all points where the angular velocity of a small paddle wheel inserted into
- Compute the flux \(\oint_{\partial \mathcal{D}} \mathbf{F} \cdot \mathbf{n} d s\) of \(\mathbf{F}(x, y)=\left\langle x^{3}, y x^{2}ightangle\) across the unit square \(\mathcal{D}\) using the vector
- Compute the flux \(\oint_{\partial \mathcal{D}} \mathbf{F} \cdot \mathbf{n} d s\) of \(\mathbf{F}(x, y)=\left\langle x^{3}+2 x, y^{3}+yightangle\) across the circle \(\mathcal{D}\) given by
- Suppose that \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) are surfaces with the same oriented boundary curve \(C\). In each case, does the condition guarantee that the flux of \(\mathbf{F}\) through
- Prove that if \(\mathbf{F}\) is a gradient vector field, then the flux of \(\operatorname{curl}(\mathbf{F})\) through a smooth surface \(\mathcal{S}\) (whether closed or not) is equal to zero.
- Verify Stokes' Theorem for \(\mathbf{F}(x, y, z)=\langle y, z-x, 0angle\) and the surface \(z=4-x^{2}-y^{2}, z \geq 0\), oriented by outward-pointing normals. THEOREM 1 Stokes' Theorem Let S be a
- Let \(\mathbf{F}(x, y, z)=\left\langle z^{2}, x+z, y^{2}ightangle\), and let \(\mathcal{S}\) be the upper half of the ellipsoid\[\frac{x^{2}}{4}+y^{2}+z^{2}=1\]oriented by outward-pointing normals.
- Use Stokes' Theorem to evaluate ∮C⟨y,z,x⟩⋅dr∮C⟨y,z,x⟩⋅dr, where CC is the curve in Figure 2. (0, 0, 1) y +2=1 S (0, 1, 0)
- Let SS be the side of the cylinder x2+y2=4,0≤z≤2x2+y2=4,0≤z≤2 (not including the top and bottom of the cylinder). Use Stokes' Theorem to compute the flux of
- Verify the Divergence Theorem for \(\mathbf{F}(x, y, z)=\langle 0,0, zangle\) and the region \(x^{2}+y^{2}+z^{2}=1\). THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W
- Use the Divergence Theorem to calculate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field and surface.\(\mathbf{F}(x, y, z)=\left\langle x y, y z, x^{2}
- Use the Divergence Theorem to calculate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field and surface.\(\mathbf{F}(x, y, z)=\left\langle x y, y z, x^{2}
- Use the Divergence Theorem to calculate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field and surface.\(\mathbf{F}(x, y, z)=\left\langle x y z+x y, \frac{1}{2}
- Use the Divergence Theorem to calculate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field and surface.\(\mathbf{F}(x, y, z)=\left\langle\sin (y z), \sqrt{x^{2}+z^{4}},
- Find the volume of a region \(\mathcal{W}\) if\[\iint_{\partial \mathcal{W}}\left\langle x+x y+z, x+3 y-\frac{1}{2} y^{2}, 4 zightangle \cdot d \mathbf{S}=16\]
- Show that the circulation of \(\mathbf{F}(x, y, z)=\left\langle x^{2}, y^{2}, z\left(x^{2}+y^{2}ight)ightangle\) around any curve \(C\) on the surface of the cone \(z^{2}=x^{2}+y^{2}\) is equal to
- Let \(\mathbf{F}\) be a vector field whose curl and divergence at the origin are\[\operatorname{curl}(\mathbf{F})(0,0,0)=\langle 2,-1,4angle, \quad \operatorname{div}(\mathbf{F})(0,0,0)=-2\]Estimate
- Let \(\mathbf{F}\) be a vector field whose curl and divergence at the origin are\[\operatorname{curl}(\mathbf{F})(0,0,0)=\langle 2,-1,4angle, \quad \operatorname{div}(\mathbf{F})(0,0,0)=-2\]Estimate
- Let \(\mathbf{F}\) be a vector field whose curl and divergence at the origin are\[\operatorname{curl}(\mathbf{F})(0,0,0)=\langle 2,-1,4angle, \quad \operatorname{div}(\mathbf{F})(0,0,0)=-2\]Suppose
- Let \(\mathbf{F}\) be a vector field whose curl and divergence at the origin are\[\operatorname{curl}(\mathbf{F})(0,0,0)=\langle 2,-1,4angle, \quad \operatorname{div}(\mathbf{F})(0,0,0)=-2\]Estimate
- The velocity vector field of a fluid (in meters per second) is\[\mathbf{F}(x, y, z)=\left\langle x^{2}+y^{2}, 0, z^{2}ightangle\]Let \(\mathcal{W}\) be the region between the
- The velocity field of a fluid (in meters per second) is\[\mathbf{F}=(3 y-4) \mathbf{i}+e^{-y(z+1)} \mathbf{j}+\left(x^{2}+y^{2}ight) \mathbf{k}\](a) Estimate the flow rate (in cubic meters per
- Let \(f(x, y)=x+\frac{x}{x^{2}+y^{2}}\). The vector field \(\mathbf{F}=abla f\) (Figure 5) provides a model in the plane of the velocity field of an incompressible, irrotational fluid flowing past a
- Figure 6 shows the vector field \(\mathbf{F}=abla f\), where\[f(x, y)=\ln \left(x^{2}+(y-1)^{2}ight)+\ln \left(x^{2}+(y+1)^{2}ight)\]which is the velocity field for the flow of a fluid with sources
- In Section 17.1, we showed that if \(C\) is a simple closed curve, oriented counterclockwise, then the area enclosed by \(C\) is given by\[\text { area enclosed by } C=\frac{1}{2} \oint_{C} x d y-y d
- Suppose that the curve \(C\) in Figure 7 has the polar equation \(r=f(\theta)\).(a) Show that \(\mathbf{r}(\theta)=\langle f(\theta) \cos \theta, f(\theta) \sin \thetaangle\) is a counterclockwise
- Prove the following generalization of Eq. (1). Let \(C\) be a simple closed curve in the plane \(\mathcal{S}\) with equation \(a x+b y+c z+d=0\) (Figure 8). Then the area of the region \(R\) enclosed
- Use the result of Exercise 39 to calculate the area of the triangle with vertices (1,0,0),(0,1,0)(1,0,0),(0,1,0), and (0,0,1)(0,0,1) as a line integral. Verify your result using geometry.Data From
- Show that \(\Phi(\theta, \phi)=(a \cos \theta \sin \phi, b \sin \theta \sin \phi, c \cos \phi)\) is a parametrization of the
- Which is the contrapositive of A⇒BA⇒B ?(a) B⇒AB⇒A(b) ¬B⇒A¬B⇒A(c) ¬B⇒¬A¬B⇒¬A(d) ¬A⇒¬B¬A⇒¬B
- Which of the choices in Question 1 is the converse of \(A \Rightarrow B\) ?Data From Question 11. Which is the contrapositive of \(A \Rightarrow B\) ?(a) \(B \Rightarrow A\)(b) \(eg B \Rightarrow
- Suppose that \(A \Rightarrow B\) is true. Which is then automatically true, the converse or the contrapositive?
- Restate as an implication: "A triangle is a polygon."
- Which is the negation of the statement, "The car and the shirt are both blue"?(a) Neither the car nor the shirt is blue.(b) The car is not blue and/or the shirt is not blue.
- Which is the contrapositive of the implication, "If the car has gas, then it will run"?(a) If the car has no gas, then it will not run.(b) If the car will not run, then it has no gas.
- State the negation.The time is 4 o'clock.
- State the negation.\(\triangle A B C\) is an isosceles triangle.
- State the negation.\(m\) and \(n\) are odd integers.
- State the negation.Either \(m\) is odd or \(n\) is odd.
- State the negation.\(x\) is a real number and \(y\) is an integer.
- State the negation.\(f\) is a linear function.
- State the contrapositive and converse.If \(m\) and \(n\) are odd integers, then \(m n\) is odd.
- State the contrapositive and converse.If today is Tuesday, then we are in Belgium.
- State the contrapositive and converse.If today is Tuesday, then we are not in Belgium.
- State the contrapositive and converse.If \(x>4\), then \(x^{2}>16\).
- State the contrapositive and converse.If \(m^{2}\) is divisible by 3 , then \(m\) is divisible by 3 .
- State the contrapositive and converse.If \(x^{2}=2\), then \(x\) is irrational.
- Give a counterexample to show that the converse of the statement is false.If \(m\) is odd, then \(2 m+1\) is also odd.
- Give a counterexample to show that the converse of the statement is false.If \(\triangle A B C\) is equilateral, then it is an isosceles triangle.
- Give a counterexample to show that the converse of the statement is false.If \(m\) is divisible by 9 and 4 , then \(m\) is divisible by 12 .
- Give a counterexample to show that the converse of the statement is false.If \(m\) is odd, then \(m^{3}-m\) is divisible by 3 .
- Determine whether the converse of the statement is false.If \(x>4\) and \(y>4\), then \(x+y>8\).
- Determine whether the converse of the statement is false.If \(x>4\), then \(x^{2}>16\).
- Determine whether the converse of the statement is false.If \(|x|>4\), then \(x^{2}>16\).
- Determine whether the converse of the statement is false.If \(m\) and \(n\) are even, then \(m n\) is even.
- State the contrapositive and converse (it is not necessary to know what these statements mean).If \(f\) and \(g\) are differentiable, then \(f g\) is differentiable.
- State the contrapositive and converse (it is not necessary to know what these statements mean).If the force field is radial and decreases as the inverse square of the distance, then all closed orbits
- The inverse of \(A \Rightarrow B\) is the implication \(eg A \Rightarrow eg B\).Which of the following is the inverse of the implication, "If she jumped in the lake, then she got wet"?(a) If she did
- The inverse of \(A \Rightarrow B\) is the implication \(eg A \Rightarrow eg B\).State the inverses of these implications:(a) If \(X\) is a mouse, then \(X\) is a rodent.(b) If you sleep late, you
- The inverse of \(A \Rightarrow B\) is the implication \(eg A \Rightarrow eg B\).Explain why the inverse is equivalent to the converse.
- The inverse of \(A \Rightarrow B\) is the implication \(eg A \Rightarrow eg B\).State the inverse of the Pythagorean Theorem. Is it true?
- Theorem 1 in Section 2.4 states the following: "If \(f\) and \(g\) are continuous functions, then \(f+g\) is continuous." Does it follow logically that if \(f\) and \(g\) are not continuous, then
- Write out a proof by contradiction for this fact: There is no smallest positive rational number. Base your proof on the fact that if \(r>0\), then \(0
- Use proof by contradiction to prove that if \(x+y>2\), then \(x>1\) or \(y>1\) (or both).
- Use proof by contradiction to show that the number is irrational.\(\sqrt{\frac{1}{2}}\)
- Use proof by contradiction to show that the number is irrational.\(\sqrt{3}\)
- Use proof by contradiction to show that the number is irrational.\(\sqrt[3]{2}\)
- Use proof by contradiction to show that the number is irrational.\(\sqrt[4]{11}\)
- An isosceles triangle is a triangle with two equal sides. The following theorem holds: If \(\Delta\) is a triangle with two equal angles, then \(\Delta\) is an isosceles triangle.(a) What is the
- Consider the following theorem: Let \(f\) be a quadratic polynomial with a positive leading coefficient. Then \(f\) has a minimum value.(a) What are the hypotheses?(b) What is the contrapositive?(c)
- Let \(a, b\), and \(c\) be the sides of a triangle and let \(\theta\) be the angle opposite \(c\). Use the Law of Cosines (Theorem 1 in Section 1.4) to prove the converse of the Pythagorean Theorem.
- Carry out the details of the following proof by contradiction that \(\sqrt{2}\) is irrational (this proof is due to R. Palais). If \(\sqrt{2}\) is rational, then \(n \sqrt{2}\) is a whole number for
- Generalize the argument of Exercise 39 to prove that \(\sqrt{A}\) is irrational if \(A\) is a whole number but not a perfect square. Choose \(n\) as before and let \(m=n
- Generalize further and show that for any whole number \(r\), the \(r\) th root \(\sqrt[r]{A}\) is irrational unless \(A\) is an \(r\) th power. Let \(x=\sqrt[r]{A}\). Show that if \(x\) is rational,
- Given a finite list of prime numbers \(p_{1}, \ldots, p_{N}\), let \(M=\) \(p_{1} \cdot p_{2} \cdots p_{N}+1\). Show that \(M\) is not divisible by any of the primes \(p_{1}, \ldots, p_{N}\). Use
- Use the Principle of Induction to prove the formula for all natural numbers \(n\).\(1+2+3+\cdots+n=\frac{n(n+1)}{2}\)
- Use the Principle of Induction to prove the formula for all natural numbers \(n\).\(1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}\)
- Use the Principle of Induction to prove the formula for all natural numbers \(n\).\(1+x+x^{2}+\cdots+x^{n}=\frac{1-x^{n+1}}{1-x}\) for any \(x eq 1\)
- Use the Principle of Induction to prove the formula for all natural numbers \(n\).\(1+x+x^{2}+\cdots+x^{n}=\frac{1-x^{n+1}}{1-x}\) for any \(x eq 1\)
- Let \(P(n)\) be the statement \(2^{n}>n\).(a) Show that \(P(1)\) is true.(b) Observe that if \(2^{n}>n\), then \(2^{n}+2^{n}>2 n\). Use this to show that if \(P(n)\) is true for \(n=k\), then