Meeting Part 1: Optimization Problems (Section 4.5)

[1] (Suggested Exercise 4.5#17, similar to book Section 4.5 Example 3 and Wed Mar 27 Class Example.) Find the points on the ellipse \(4x^2+y^2=4\) that are farthest away from the point \((1,0)\) (You must use calculus and show all details clearly. No credit for just guessing values.)

[2] (Suggested Exercise 4.5#22, similar to book Section 4.5 Example 5 and Fri Mar 29 Class Example.) Find the area of the largest rectange that can be inscribed in a right triangle with legs of lengths 3cm and 2cm if two sides of the rectangle lie along the legs. (You must use calculus and show all details clearly. No credit for just guessing values.)

[3] (Suggested Exercise 4.5#11) If 1200 cm ² of material is available to make a box with a square base and an open top, find the largest possible volume of the box. (You must use calculus and show all details clearly. No credit for just guessing values.)

Meeting Part 2: Newton's Method (Section 4.6)

[4] Students work in pairs on this ( Class Drill on Using Newton's Method )

If there is time remaining: Meeting Part 3: Challenge Problem (Section 4.5)

[5] (Suggested Exercise 4.5#39) Find an equation of the line through the point \((3,5)\) that cuts off the least area from the first quadrant. (You must use calculus and show all details clearly. No credit for just guessing values.)
Hint for an outline of the solution:

• Consider a line \(L\) that has axis intercepts at \((a,0)\) and \((0,b)\), where \(a,b\) are unknown positive real numbers. Sketch the line \(L\), and compute the area \(A\) that the line cuts off in the first quadrant. Your expression for area \(A\) should involve the two unknowns \(a,b\). With just this expression for \(A\), it would be impossible to minimize \(A\).
• Now, satisfy the additional requirement that the line \(L\) must pass through the point \((3,5)\).
• You now have two equations involving the constants \(a,b\). The first is an equation involving \(A,a,b\). The second is an equation just involving \(a,b\). Solve the second equation for \(b\) and use that result to eliminate \(b\) in the first equation. The result should be a single equation involving \(A,a\).
• Using calculus , find the value of \(a\) that minimizes \(A\). (Remember that \(a\) must be positive .)
• Find the corresponding value of \(b\).
• Find the slope \(m\) of line \(L\).
• Find the equation of line \(L\).

Meeting Part 1: Antiderivatives Satisfying an Extra Condition (Section 4.7)

[1] (Suggested Exercise 4.7 #15) Let $$f(x)=7x-3x^5$$

1. Find the General Antiderivative , \(F(x)\).
2. Find the Particular Antiderivative that satisfies \(F(1)=5\).

[2] (not like a book exercise) Let $$f(t)=3e^t-4$$

1. Find the General Antiderivative , \(F(t)\).
2. Find the Particular Antiderivative that satisfies \(F(0)=8\).

[3] (review of prerequisites) Draw the first quadrant of the unit circle , with important famous angles \(\theta=0,\pi/6,\pi/4,\pi/3,\pi/2 \) shown, along with the \((x,y)\) coordinates of the points where the rays of those angles intersect the circle. You need to know these angles and the values of the trig functions at these angles!

[4] (4.7#27) Find \(f(t)\) such that $$f'(t)=10\cos t - \sec^2 t \ \text{ for } \ -\pi/2 \lt t \lt \pi/2 \ \text{ and that } \ f(\pi/3)=13$$

[5] (4.7#20) Suppose that $$f''(x)=30x-\sin x$$ Find \(f(x)\).

Meeting Part 2: Problems about Position , Velocity , and Acceleration (Section 4.7)

Remember that for an object moving in one dimension, the velocity , \(v(t)\), is the derivative of the position , \(s(t)\). That is, $$s'(t) = v(t)$$

Therefore, position , \(s(t)\), is an antiderivative of the velocity , \(v(t)\).

Also remember that the acceleration , \(a(t)\), is the derivative of the velocity , \(v(t)\).

Therefore, velocity , \(v(t)\), is an antiderivative of the acceleration , \(a(t)\).

Furthermore, recall that when an object falls freely under the influence of gravity , it is known that the object will have constant acceleration with a value $$a=-32 \ \text{ft/s}^2$$ The negative sign may be confusing. The reason for the negative sign is that the positive position direction is up . Since gravity makes objects fall down , it is acclerating them in the negative position direction. Hence, the acceleration gets a negative sign.

[6] (based on 4.7#40, similar to 4.7#47) Suppose that an object is moving in one dimension with velocity $$v(t)=9\sqrt{t} \ \text{ ft/s}$$

1. Find the general form of the position function , \(s(t)\). That is, find the General Antiderivative of \(v(t)\), but instead of calling it \(V(t)\), call it \(s(t)\).
2. Suppose that it is also known that the initial position is \(s(0)=13 \ \text{ft}\). Find the position function . That is, find the Particular Antiderivative that satisfies \(s(0)=13\).

[7] (based on 4.7#43) Suppose that an stone is dropped off a tower that is 400 feet tall and falls freely. Let position be defined to be zero at ground level , and remember that the positive position direction is up .

1. What is the value of the initial position of the stone, \(s(0)\)?
2. What is the value of the initial velocity of the stone, \(v(0)\)?
3. The acceleration of the stone is constant. What is the value of the the acceleration , \(a\)?
4. Given that the velocity , \(v(t)\), is an antiderivative of the acceleration , find the general form of the velocity function , \(v(t)\). That is, find the General Antiderivative of \(a(t)\), but instead of calling it \(A(t)\), call it \(v(t)\).
5. Knowing what you know about the initial velocity , \(v(0)\), find the particular form of the velocity function , \(v(t)\). That is, find the Particular Antiderivative of \(a(t)\) that satisfies $$v(0) = \text{ initial velocity that you identified earlier}$$
6. Given that the position , \(s(t)\), is an antiderivative of the velocity , find the general form of the position function , \(s(t)\). That is, find the General Antiderivative of \(v(t)\), but instead of calling it \(V(t)\), call it \(s(t)\).
7. Knowing what you know about the initial position , \(s(0)\), find the particular form of the position function , \(s(t)\). That is, find the Particular Antiderivative of \(v(t)\) that satisfies $$s(0) = \text{ initial position that you identified earlier}$$ The formula that you have found for the position function , \(s(t)\) gives the position of the stone above ground level at time \(t\).
8. What is the time when the stone reaches ground level?
9. What is the speed of the stone when it strike the ground?

Meeting Part 3: Problems about l�Hospital�s Rule (Section 3.7)

You might not remember that l�Hospital�s Rule was covered in the last class meeting before Exam X2, so there were no problem about l�Hospital�s Rule on Exam X2. At the time, I said that there would be problems about l�Hospital�s Rule on Exam X3. Well, here we are.

[8] (Suggested Exercise 3.7#1) Consider the following limit, which is in \(\frac{0}{0}\) indeterminate form : $$\lim_{x\rightarrow 1}\frac{x^2-1}{x^2-x}$$

1. Find the limit using techniques from Chapter 1.
2. Find the limit using l�Hospital�s Rule .

[9] (Suggested Exercise 3.7#3) The following limit is in \(\frac{0}{0}\) indeterminate form . Find the limit using l�Hospital�s Rule $$\lim_{x\rightarrow \left(\pi/2\right)^+}\frac{\cos x}{1-\sin x}$$

[10] (Suggested Exercise 3.7#25) The following limit is in \(\infty \cdot 0\) indeterminate form . Find the limit using l�Hospital�s Rule $$\lim_{x\rightarrow \infty}x^3e^{-x^2}$$ Hint: Rewrite the product as a quotient, so that it will then be in \(\frac{0}{0}\) indeterminate form . Then use l�Hospital�s Rule .

Recitation Part 1: Basic Problems Using the Evaluation Theorem

Recall The Evaluation Theorem (ET) as presented in the book and in lecture on Mon Apr 15, using the terminology of antiderivatives :

(the relationship between definite integrals and antiderivatives )

If \(f(x)\) is continuous on the interval \([a,b]\), then $$\int_a^bf(x)dx\underset{\text{ET}}{=}F(b)-F(a)$$ where \(F(x)\) is any antiderivative of \(f(x)\).

[1]: (5.3#3)$$\int_{-2}^{0}\left(\frac{1}{2}t^4+\frac{1}{4}t^3-t\right)dt$$

[2]: (5.3#13)$$\int_{1}^{2}\left(\frac{x}{2}-\frac{2}{x}\right)dx$$

[3]: (5.3#7)$$\int_{0}^{\pi}\left(5e^x+3\sin x\right)dx$$

Recitation Part 2: Using the Net Change Theorem

Recall The Net Change Theorem (NCT) as presented in the book and in lecture on Mon Apr 15, using the terminology of derivatives :

(the integral of a rate of change of a quantity is the net change of that quantity )

If \(F(x)\) is differentiable on the interval \([a,b]\), then $$\int_a^bF'(x)dx\underset{\text{NCT}}{=}F(b)-F(a)$$

[4]: (5.3#59) An object moves along a line with velocity $$v(t)=3t-5 \ \text{ for } \ 0\leq t \leq 3$$ where \(t\) is the time in seconds and \(v(t)\) is the velocity at time \(t\), in meters per second.

1. Find the displacement of the object during the time interval. Give an exact answer , with correct units.
2. Illustrate your result for (a) using a graph of the velocity \(v(t)\).
3. Find the distance traveled by the object during the time interval. Give an exact answer and a decimal approximation , rounded to 3 decimal places.

[5]: (5.3#60) An object moves along a line with velocity $$v(t)=t^2-2t-3\ \text{ for } \ 0\leq t \leq 6$$ where \(t\) is the time in seconds and \(v(t)\) is the velocity at time \(t\), in meters per second.

1. Find the displacement of the object during the time interval \([2,5]\). Give an exact answer , with correct units.
2. Illustrate your result for (a) using a graph of the velocity \(v(t)\).
3. Find the distance traveled by the object during the time interval \([2,5]\). Give an exact answer and a decimal approximation , rounded to 3 decimal places.

Recitation Part 3: Harder Problems Using the Evaluation Theorem

[6]: (5.3#18) Hint: The bad news is that we have no antiderivative rules that work on a function as complicated as the function that is the integrand here. Your only hope is to rewrite the integrand in a form for which our basic antiderivative rules do work. The good news is that the integrand can be simplified a lot! $$\int_{0}^{\pi/3}\left(\frac{\sin \theta +\sin \theta \tan^2 \theta}{\sec^2 \theta}\right)d\theta$$

[7]: (5.3#9) Hint: You�ll have to rewrite the integrand as a sum of power functions before integrating. $$\int_{1}^{4}\left(\frac{4+6u}{\sqrt{u}}\right)du$$

[8]: (5.3#23) Hint: You�ll have to rewrite the integrand as a sum of power functions and \(\frac{1}{x}\) functions before integrating. $$\int_{1}^{e}\left(\frac{x^2+x+1}{x}\right)dx$$

Recitation Part 3: Some More Hard Problems Using the Evaluation Theorem

(You had some hard problems in last Tuesday�s Recitation.)

Recall The Evaluation Theorem (ET) as presented in the book and in lecture on Mon Apr 15, using the terminology of antiderivatives :

(the relationship between definite integrals and antiderivatives )

If \(f(x)\) is continuous on the interval \([a,b]\), then $$\int_a^bf(x)dx\underset{\text{ET}}{=}F(b)-F(a)$$ where \(F(x)\) is any antiderivative of \(f(x)\).

[1]: (5.3#11) Hint: You�ll have to rewrite the integrand as a sum of power functions before integrating. $$\int_{0}^{1}x\left(\sqrt[3]{x}+\sqrt[4]{x}\right)dx$$ Give an exact answer and a decimal approximation , rounded to 3 decimal places.

[2]: (5.3#15) Hint: You�ll have to do some sleuth work to figure out one of the antiderivatives. Try checking your book in Section 5.3. $$\int_{0}^{1}\left(x^{10}+10^x\right)dx$$ Give an exact answer and a decimal approximation , rounded to 3 decimal places.

[3]: (5.3#29) Hint: Remember that the function \(|x|\) is a piecewise-defined function. That is, the formula for \(|x|\) depends on which piece of the domain that you are in. That will mean that you will need to break up this definite integral on the interval \([-1,2]\) into two definite integrals, each on a smaller interval. $$\int_{-1}^{2}\left(x-2|x|\right)dx$$ Give an exact answer .

Recitation Part 12: Using the Fundamental Theorem of Calculus, Part 1

Recall the Fundamental Theorem of Calculus, Part 1

If \(f\) is continuous on the interval \([a,b]\), then $$\frac{d}{dx}\left(\int_a^xf(t)dt\right)\underset{\text{FTC1}}{=}f(x) \text{ for } \ a \lt x \lt b$$

[4]: (5.4#6) The function \(g(x)\) is defined by the integral: $$g(x)=\int_{3}^{x}e^{t^2-t} \ dt$$ Find \(g'(x)\).

[5]: (5.4#10) The function \(g(x)\) is defined by the integral: $$g(x)=\int_{0}^{x}\sqrt{1+\sqrt{t}} \ dt$$ Find \(g'(x)\).

[6]: (5.4#10) The function \(h(x)\) is defined by the integral: $$h(x)=\int_{0}^{\tan x}\sqrt{1+\sqrt{t}} \ dt$$ (Hint: You will need the Chain Rule .)

Recitation Part 3: Computing the Average Value of a Function on an Interval

Recall the Definition of the Average Value of a Function on an Interval

If \(f(x)\) is continuous on the interval \([a,b]\), then the Average Value of \(f(x)\) on the interval \([a,b]\) is defined to be the number

[7]: Find the average value of the function \(f(x)= \frac{1}{x}\) on the interval \([1,4]\). Simplify your answer.

[8]: Find the average value of the function \(f(x)= \sin (x) \) on the interval \([0,\pi]\). Simplify your answer.

[9]: Find the average value of the function \(f(x)= \sec^2(\theta)\) on the interval \([0,\pi/4]\). Simplify your answer.