Calculus
Questions & Answers
1) A gas station sits at the intersection of a north-south road and an east-west road. A police car is traveling toward the gas station from the east, chasing a stolen truck which is traveling north away from the gas station. The speed of the police car is 100 mph at the moment it is 3 miles from the gas station. At the same time, the truck is 4 miles from the gas station going 80 mph. What is the rate of change of the straight line distance between the two vehicles at this moment?
y" - 4y + 4y = 0; y(0) = 2,y' (0) = 1 Use Laplace Transformation to solve it. Do not use any other method. show all steps, copy the part of Laplace table that you are using.
(5) The number of deer in a National Park is modeled by P(t), where time t is given in years. 1070€0.23t 0.9 + €0.23t P(t) Use the Intermediate Value Theorem to show that the model predicts there will be exactly 1000 deer at some time t, where 10 < t < 15. You do not need to find the exact value of t that gives 1000 deer.
Verify § ².d ²³ = SS (²x²). 1³² с that is show that the LHS =RHS, for in the surface The heat Z ph I deform, Stokes Theoran Ĵ 270 S: x² + y² +2²=3 =>0 trick that I use 골 10~ S31 7²x TX 7²x Imagine a bubble on a CIRCULAR rim C₁, Here &c, closed curve = §₁²₂² = § ²3₂ = § ₁₂₁ For a vector fuld we will find different surface SS (0x²) ds₁ = ²d² | C₁=C₁=C3=C4/ & 215 1₁² That for each the bubble so that 11 =12² in (14) only x Si 6₂ (0x²) Sf₂ (0x²).ds₂ = Fid²³, SS (oxf) ds = ) & F.d have to calculate the area of a circle!/nSo by luoking at my diagrams you can appreciate for each unusual Irregular smooth /bubble surface S₁, S₂ and 53 a) ff (√x²).d³₁ = √ ².dr SI C₁ 1) $ (0ײ).d³₂ = § ³.4² = & Fd² S2 2 =) ff ( x ²).d.² = & F₁d² = §£ F.d² CI S3 VX² I have F.d d d) $ (x²)² = & fid² = 6.² Ju JC4 a) = b) = c) = d) S4 but in of F. d² the bubble has been flattened planar surface and 1=1 here ₁ to a So f 11 where EASIEST TO SOLVE Alix да C4 î = F₁T +2₂3 +5h => (PXF) onds = (F₁, F₂, F) 12 => SS (Gx²³²) •nds = SS F3 ds F3 ds == ²0 and integral is easier. X
9 Find the following indefinite or definite integrals. (a) (4 points) (3r5 −1+) dx I (b) (4 points) f/4 sec²(x) dx (c) (6 points) Find the indefinite integral using the substitution method. [(3x²- +2) cos(x³ + 2x + 1) dr
8 (10 points) (a) Find the critical numbers of the function f(x) = x³-3x+5 in the interval [-2,3]. Then classify them to be local maximum and the local minimum of f(x) on the interval [-2,3] (b) Find the open intervals where the function is concave up or concave down. Also, find the inflection point.
4. How accurate is this approximation? Use the Remainder Estimation Theorem to determine the upper bound for 102/3 - P₂(10)| = |R₂(10)|.
(6a) (4 points) Differentiate the function f(r) = ³ sec(r). (6b) (4 points) Differentiate the function f(x) = 2x²+1 (6c) (4 points) Differentiate the function g(x) = [e+tan(z)]7. (6d) (4 points) Differentiate the function (x) = ln(x²+x+10).
Problem: Let f(x) = x²/3. In Math 151, we could use the linearization of f centered at a = 8 to find an approximation to 102/3. In Math 152, we will use a second-order Taylor polynomial to provide both a calculation and the Remainder Estimation Theorem to give an upper bound on the possible error this approximation has. 1. Find the second order Taylor polynomial P₂(x) centered at a = 8.
