RULING AND SOLVING! 1. Review the basic concepts of Basic Rules of Differentiation. 2. Solve the given function using chain rule. 3. Show you complete SOLUTION AND FINAL ANSWER on the given functions. 1. f(x) = (6x² - 2x+1)* 2. ƒ (x) = (3x² − 4) ²³ 3 3. f(x)=(5x² +8x)³ 1 4. f(x)= (7x³-5x-2)³ 5.f(x)=√√5x² +2
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- 2.1 Can you show me the steps to solve this problem? 27) If f(x) = 3x^2 - x^3, find f'(1) and use it to find an equation of the tangent line to the curve y = 3x^2 - x^3 at the point (1,2).Q1) Use the technique of logarithmic differentiation to differentiate the function F(X)=[(X+1)^3*SQR(3X-1)]/(X^2+16)^6 Q2)Determine the linearization of the functionf (X ) = SQR (4X+ 1)at x=6. Use that linearization to approximate the value ofSQR (25.2) . Q3) A rocket is blasting off from a launchpad and you are going to film the takeoff. You position yourself at a minimum safe distance of 4000 feet away. As the rocket launches straight up at1000 feet per second, you rotate the camera to keep the rocket in the frame. Three seconds after thelaunch, determine both the rate at which the distance between you and the rocket is increasing, as wellas the rate at which the angle your camera’s line of sight creates with the horizontal is increasing. Ifneeded, round both measurements to the nearest 0.001 and include units.1. A room with a volume of 5000 ft3 is full of air which contains 2% (CO). Pure air containing no (CO) is pumped into the room at the rate of 200 cubic feet per minute, and the well-mixed air is vented from the room at the same rate. a. Find the equation that models the amount of (CO), x(t), in ft3, which is in the room, as a function of time. b. How long will it take for the concentration of (CO) in the room to reach 0.01%?
- When a species of pigs is introduced onto an island, its population increases 15% each year. This population t years after introduction is given by P = k * a^t where k and a are constants and t is greater than or equal to 0 what is the value of k and what does it represent in context of the problem?Solve the following problems below about derivative. Simplify your answers if necessary. 1. A three-sided fence is to be built next to a straight section of a river, which forms the fourth side of the rectangular region. The enclosed area is equal to 1800 square feet. Find the maximum perimeter and the dimension of the corresponding enclosure. 2. Suppose two feet long wire is to be cut into two pieces,with each piece to be formed into a square. Find the size of each piece to maximize the total area of the two squares. 3. A spherical snowball is made so that its volume is increasing at a rate of 8 cubic feet per minute. Find the rate which the radius is increasing when the snowball is 4 feet in diameterJuaquim's starting salary at a company is $42,000 per year. On average, employees receive a 6.5% increase in salary every year. Which equation shows what his salary will be in x years? A) s=(1+0.065)xs=(1+0.065)x A), s is equal to open paren 1 plus 0 point 0 6 5 close paren to the x th power B) s=42000(1+0.065)xs=42000(1+0.065)x B), s is equal to 42000 times open paren 1 plus 0 point 0 6 5 close paren to the x th power C) s=42000(0.065)xs=42000(0.065)x C), s is equal to 42000 times 0 point 0 6 5 to the x th power D) s=42000(1−0.065)x
- The derivative of a function F(x) is f(x)=4x3. If F(x)= 10, what is the constant of integration? a. 10 b. 6 c. -1 d.-6Rod says that he is thinking of two functions that have the following characteristics:- One is rational, and has a y-intercept at -2- One is trigonometric, and does not include the cosine function- One contains the digit “3” and the other does not.- Both have an instantaneous rate of change of 1.23 at ? = 2- The two functions intersect at ? = 2provide one example of a pair of functions that fulfill this. Explain your thoughtprocess, a screenshot and calculations to prove it.2.6 Can you show me the steps to solve the problem? Directions: Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. 23) x4y2 - x3y + 2xy3 = 0
- Use differential approximations in the following problem. A company will sell N units of a product after spending $x thousand in advertising, as given by N=60x-x^2, 5<=x<=30. Approximately what increase in sales will result by increasing the advertising budget from $5,000 to $6,000? From $20,000 to $21,0001. A variant of a new virus is said to grows infection of a human population according to the function f(t) = 300e0.02t where t is measured in days. (a) How many are expected to get infected after a month (30 days) ? (b) How long will it take before it will ever reach an infected population to 1,000?11 .Consider the function f(x)=2x^2 +(−4)x What is the value of the constant of integration such that the primitive of this function passes through the point (4,3)? Select one: a. -7.67 b. 13.67 c. -61.00 d. -9.00