3. x(t) = e kt cos(ln(t)) 4. G(s) = s2 k 2 + s 2

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1 M7 Fall 28: Final Examination Practice Problems Section and the rest. You can check your answers in WebWork. Full solutions in WW available Tuesday evening 2/. Problem. Compute the derivative of the given function:. f(θ) = cos(2θ 2 + θ + 2) 2. g(u) = ln ( sin 2 (u) ) 3. r(y) = arctan(y 3 + ) 4. s(t) = arcsin(βt) 5. r(θ) = θ 3 sin(θ) 6. s(t) = tan(t) + sec(t) Problem 2. Compute the derivative of the given function:. f(x) = xe x cos(x) 2. g(x) = ex tan(x) 3. x(t) = e kt cos(ln(t)) 4. G(s) = s2 k 2 + s 2 Problem 3. Which of the following is an antiderivative for the function f(x) = e x sin(e x ). Circle all the correct answers.. F (x) = cos(e x ) 2. F (x) = sin(e x ) 3. F (x) = x et cos(e t ) dt 4. F (x) = e x cos(t) dt 5. F (x) = x et sin(e t ) dt 6. F (x) = e x sin(t) dt Problem e 9 x x x x 5 cos(2t + )dt Compute the following integrals: x 2 x( + x) 6 e kx ( + t 2 ) dt Problem 5. Compute the following integrals: ln(2) π e s+ ds e x sec 2 (t/4) dt xe x π/4 x( + x) 6 sin(2x) cos(2x) 3x x π 4 cos(x) Problem 6. Suppose that f (x) > for x near a point a. Then the linearization of f at a is. an over approximation 2. an under approximation 3. unknown without more information.

2 2 Problem 7. Given F (x) = 2 + π x e t2 dt find the linearization of F at a =, L(x) =? Problem 8. Energy lost from the earth due to radiation into space depends on the current temperature of the Earth T, and is approximated as E = 4πr 2 ɛσt 4 where ɛ is the emissivity of the Earth s atmosphere, which represents the Earth s tendency to emit radiation energy. This constant depends on cloud cover, water vapor as well as greenhouse gas concentration in the atmosphere, such as carbon dioxide and methane levels. The parameter σ is a physical constant called the Stephan-Boltzmann constant. Without solving for T as a function E, find a formula for dt de. (Hint: differentiate both side of the equation with respect to E.) Problem 9. For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v 2. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current with speed s (where s < v), then the time required to swim a distance L is L/(v s), and the total energy E required to swim the distance is given by v2 E(v) = al, for v > s. (v s) Here a, s and L are positive constants. Find the critical point(s) of E(v) in terms of these constants. Does v has a local max or min at v = 2s? Problem. A popcorn box was created from a 2 2 square with the finished box shown on the right. a. Express the volume of the box, V, as a function of x. b. What is value of x maximizes the Volume? x/2 2 x 6 x x

3 3 Problem. Water pours into a conical tank of height m and radius 5 m at a rate of 6 m 3 /min. The Volume of a conical tank of radius r and height h is given by V = π 3 r2 h. At what rate is the water level rising when the level is 4 m high? 5m r h m Problem 2. The planet Quirk is flat. The satellite in the figure travels above the planet. The radar tracks the satellite and computes the distance r, the angle θ as functions of time and then numerical computes θ and r. a. In terms of θ, θ, r, and r find the ground speed x of the satellite. b. What is the ground speed, x, when θ = π/2, θ = 2 rad/sec, r = km and r = km/sec? Problem 3. As Claudia walks away from a 2.64 m lamppost, the tip of her shadow moves twice as fast as she does. What is Claudias height? Problem 4. First find x then find x if x (t) =, x () = and x() = 6. Problem 5. Below is pictured the graph of the function f(x), its derivative f (x), and an antiderivative f(x). Identify f(x), f (x) and f(x).

4 4 Problem 6. Let A(x) = x f(t)dt, with f(x) as in the figure below. A (x) at x = P is: 2. Where does A(x) have a local minimum? 3. Where does A(x) have a local maximum? 4. A(x) < for all x between and P, True or false? Problem 7. When you slice a loaf of bread, you change its volume. Let x be the length of the loaf from one end to the place where you cut off the last slice. Let V (x) be the volume of the loaf of length x (see figure). For each x, dv is. the volume of a slice of bread. 2. the area of the cut surface of the loaf where the last slice was removed. 3. the volume of the last slice divided by the thickness of the slice. Problem 8. a. Given the graph of y = +x, indicate on the graph below the rectangles used to calculate the right endpoint approximation to the integral b. Calculate R x. with three subintervals (R 3 ) c. Will this estimate be larger or smaller than the actual value of definite integral? Circle one: Smaller Larger 2 3

5 5 Problem 9. a. Suppose that g(x) is differentiable for all x and that g (x) 3 for all x. Assume also that g() = 4. Based on this information, use the Mean Value Theorem to determine the largest and smallest possible values for g(4). b. Note it follows from the FTC- that So if g (x) 3 for all x then g(4) g() = 4 4 g (x). g (x) (fill in the blanks). Show that you get the same estimate you got in part (a) above using the comparison theorem for integrals instead of the MVT. Problem 2. Evaluate the following integrals given the graph of f and that the two parts of the graph below are semicircles. (a) (b) (c) (d) f(x) = f(x) = f(x) = f(x) = Problem 2. Even before the invention of calculus, Archimedes using an ingenious method, discovered that the area A under a parabolic arch is two-thirds the base b times the height h: A = 2 3 bh Use calculus to verify by finding the area under the curve y = h( 4 x 2 ) for b/2 x b/2. b 2 Problem 22. ( d x 4 a. b. 2x Find the following derivatives: ) sin(t) t dt d 2x ) (x 2 sin(t) dt t c. d. d 2x sin(t) dt t. x 2 ) d x 5 (x 5 sin(t) dt. 2 t

6 6 Problem 23. defined by Give the interval(s) for which the function F is increasing. The function F is F (x) = x t 3 t 2 + dt Problem 24. [Multiple choice] Water is pouring out of a pipe at the rate of f(t) gallons/minute. You collect the water that flows from the pipe between t = 2 and t = 4. The amount of water you collect can be represented by:. 4 2 f(x) 2. f(4) f(2) 3. (4 2)f(4) 4. the average of f(4) and f(2) times the amount of time that elapsed Problem 25. An ant moves in a straight line with the velocity v(t) = cos(t) (m/s). Find the displacement and distance traveled over the time interval [, 3π] seconds. Problem 26. The height, h at time t of a fluid in a cylindrical water tank above a hole in the bottom is given by ( a ) 2 h(t) = H 2 t for t 2 H a where a is a parameter that depends on the radius of the tank, the radius of the hole in the tank as well as the type of fluid in the tank. H is the initial height of the fluid before the plug is pulled. Find the rate at which the volume of fluid is changing as a function of time. Recall, V = πr 2 h. h(t) Problem 27. If the units for t are hours and the units of Q are gallons per hour, then the units for 4 Q(t)dt are. Problem 28. As Water flows through a tube of radius R under laminar flow the velocity v down the tube only depends only on the distance r from the center of the tube. The velocity satisfies dv/dr = kr where k is a constant and at the walls of the tube is zero so v(r) =. Determine v(r). Problem 29. Let F (x) = Problem 3. x Find f given that sin 2 (t) dt. Evaluate the limit F (x) lim x x 2. f (x) = sin(x) sec(x) tan(x), f(π) =. Problem 3. Suppose s is a twice differentiable function with the following information given: s (t) = 2, s () =, s() =. First find s (t) and then find s(t). t 2