umich calc 17 [Notes & HW Answers] 5.1 How Do We Measure Distance Traveled? [Prepwork 5.1] Q1. Consider the velocity data for the car shown in table 5.1. Find lower and upper estimates for the distance it travels between 𝑡=4 and 𝑡=8. A: lower = 164 upper = 184 Write the difference between these as the following product: difference = (Δ𝑡)(Δ𝑣)= (2) (10) Q2. The velocity of a bicycle is given by v(t) = 4t feet per second, where t is the number of seconds after the bike sta.. 2022. 12. 6. [Notes & HW Answers] 4.3 Optimization and Modeling [Prepwork 4.3] Q1. Consider the following problem: You are standing on a river bank and want to get to a house that is on the other side of the river and 800 feet up the river as quickly as possible. The river is 100 feet wide. You can walk along the river at a speed of 5 ft/sec or swim across the river at a speed of 3 ft/sec. What path should you take? A: Which of the following quantities are y.. 2022. 12. 6. [Notes & HW Answers] 4.1 Using First and Second Derivatives [Prepwork 4.1] Q1. Suppose f(x)f(x) is a continuous function that has a local maximum at the point (3,2).Which of the following statements must be true? A: (1) If a is just a little greater than 3, then f(a) ≤ 2. (2) f(x) has a critical point at x=3. (3) If a is just a little less than 3, then f(a) ≤ 2. Q2. Suppose f(x)f(x) is a continuous function that has a local minimum at (1,6)(1,6) and no o.. 2022. 12. 6. [Notes & HW Answers] 3.7 Implicit Functions [Prepwork 3.7] Q1. Which of the following points satisfies the equation 𝑥^2−𝑥𝑦^2=2? A: (1) (-2^(1/2)/2, (4.5)^(1/4)) ; (2) (-2^(1/2),0) Q2. (1) Find dy/dx when x^2 - xy^2 =2. A: (1) dy/dx = (y^2-2x)/-2xy (2) Find an equation for the tangent line to x^2 - xy^2 = 2 at the point (2,1). A: (2) y = (3/4)(x-2) + 1 Q3. Find all points satisfying the equation 𝑥2−𝑥𝑦2=2x2−xy2=2 where the tangent line is v.. 2022. 11. 21. [Notes & HW Answers] 3.6 The Chain Rule and Inverse Functions [Prepwork 3.6] Q1. Let 𝑔(𝑥)g(x) be an invertible, differentiable function with values given in the table below. 𝑥 2 5 8 𝑔(𝑥) 8 2 0 𝑔′(𝑥) -2 -0.25 -0.1 Find a formula for the tangent line of 𝑔^(−1)(𝑥) at 𝑥=2. A: y = -4(x-2)+5 Q2. Find the derivative of 𝑓(𝑥)=arctan(5ln(𝑥)). A: f'(x) = 5/(x*(25*ln^2(x)+1) [HW 3.6] Q1. Find the derivative of the function f(t), below. f(t) = ln(t^9 +8) A: f'(t) = 9t^.. 2022. 11. 21. [Notes & HW Answers] 3.5 The Trigonometric Functions [Prepwork 3.5] Q1. The Bay of Monterey in California is known for extreme tides. The depth of the water, y, in meters can be modeled as a function of time, t, in half-hours after midnight, by y=12+6cos(t). How quickly is the depth of the water rising or falling at 3 a.m.? (Make sure you compute this in radians and give your answers to three decimal places.) Rising at (in m/half-hours) A: 1.676 Q.. 2022. 11. 21. [Notes & HW Answers] 3.4 The Chain Rule [Prepwork 3.4] Q1. The length,L, in micrometers (𝜇μm), of steel depends on the air temperature degrees celsius, and the temperature depends on time,t, measured in hours. If the length of a steel bridge increases by 0.25μm for every degree increase in temperature, and the temperature is increasing at 4 degrees celsius per hour, how fast is the length of the bridge increasing? A: At a rate of 1 μm.. 2022. 11. 21. 이전 1 다음
