Basic Math question and answers for August 25, 2024
- Q Find a formula for f(x) if f(x) =[x+1/ x+1]^3-(x)=0(-)'(x) =Hint: Undo the operations of f(x) from the outside in. Undo the cubed operation first.
- Q Given the following functions: f(u) = u7/2 and g(z) = x8 +1. Find:f(g(x)) =f'(u) =f'(g(x)) =g'(x) =(og)'(x) =
- Q Consider the function f(x), for which f(0) = 5 and f'(0) = 11.Find h'(0) for the function h(x) = 1/ f(x)h' (0) =
- Q If (x) = (3x ? 4) (6x + 6) then:'(x) =[Suggestion: use logarithmic differentiation.]
- Q Given that f(z) = 2x+cos(x) is one-to-one, use the formula (f )'(x) = 1/ f'(f (x))to find (f )'(1).()'() =
- Q The slope of the line tangent to the curve 4x + 3xy + 2y = 44 at the point (2, 2) is:
- Q All you know about a differentiable function f(x) is that:'(3) = f'(4) = f'(17) = 0and that:f'(x) > 0 for x < 3f'(x) < 0 for 3 < x <...
- Q If f(x) = x5 h(x) for some function h(x) and you know that:h(-3) = -8h'(-3) = - 5then f'(-3) =
- Q The critical numbers for the functionf(x) = 2x-21x + 36x - 7 are:-
- Q Find the average rate of change of h(x) = 2x-7x from x = 1 to x = 5.
- Q The circumference of a sphere was measured to be 89 cmwith a possible error of 0.5 cm. Use linear approximation toestimate the maximum error in the calculated surface area.Estimate the...
- Q A manufacturer is making cylindrical cans that hold 300 cm. Thedimensions of the can are not mandated, so to save manufacturing costs,the manufacturer wants to make the can that uses...
- Q Ith (x) = x + 16x 10 on the interval [-4, 6]:Apply the mean value theorem to find the value of C in the interval(-4, 6) such that h' (c)...
- Q A plane traveled 1120 miles to Cheyenne and back. The trip there was with the wind. It took 10 hours. The trip back was into the wind. The trip back...
- Q An arrow is propelled upward. It is s(t) = - 16t + 80t + 490 feet off the ground after 1 seconds. Find the velocity of the arrow at 3...
- Q Find all numbers c that satisfy the conclusion of Rolle's Theorem for the following function.If there are multiple values, separate them with commas; enter N if there are no such...
- Q Consider the function f(x)=2x - 6x 90x+6 on the interval -4,9] Find the average or mean slope of the function on this interval By the Mean Value Theorem, we know...
- Q Suppose y = sinx + 2sinx. Find dy/dx.Answer:
- Q The radius of a cylinder is increasing at a rate of 1 meter per hour, and the height of the clinder is decreasing at a rate of 4 meters per...
- Q Show that the function f(x)=x +8x+2 has exactly one zero in the interval [-1.0].Which theorem can be used to determine whether a function f(x) has any zeros in a given...
- Q The length of a rectangle is increasing at a rate of 9 cm/sec while the width w is decreasing at a rate of 9 cm/sec. When t= 8 cm and...
- Q A balloon, which always remains spherical, has a variable radius. If the radius is increasing at the rate of 3 units/sec then find the rate at which its volume is...
- Q Which sequence of transformations will yield the graph of g(x) = 2^x +8 + 1Horizontal shift 1 units to the left; Vertical shift 8 units upHorizontal shift 8 units to...
- Q Select the best choice for the definition of average rate of change.A. The rate at which a function's y-values (output) are changing as compared to the function's x-values (input).B. The...
- Q The equation y = 664(1.031)t models the number of students in a certain state that go to out of state colleges/universities each year from 1992 through 1999, where y is...
- Q A fast food restaurant estimates that the mean sodium content in one of its breakfast sandwiches is 940 milligrams. A random sample of 44 breakfast sandwiches had a mean sodium...
- Q Calculate the derivative of the following function.y = sin (4 cos x)
- Q A theorem states that one local extremum implies absolute extremum. Verify that the following function satisfies the conditions of this theorem on its domain. Then find the location and value...
- Q Suppose that the total profit (in dollars) from the sale of xskateboardsis given by P(x) = 30x - 0.6x - 240 Determine themarginalaverage profit to approximate the profit from the...
- Q The functions=t-15t +75t,0?t? 5, gives the position of a body moving on a coordinate line, with s in meters and t in seconds.a. Find the body's displacement and average velocity...
- Q Calculate the derivative of y with respect to , if xy + 3xy = x + y.dy/dx=
- Q The radius of a sphere is measured as 9 centimeters, with a possible error of 0.025 centimeter.(a) Use differentials to approximate the possible propagated error, in cm, in computing the...
- Q Calculate dcos(u)/dx for the following choices of u(x) :u(x) = 6 x, dcos(u(x))/dxu(x) = 1/x, dcos(u(x))/dxu(x) = tan(x), dcos(u(x))/dx
- Q Find a, b, c, d such that the cubic function f(x) = ax + bx + cx+d has horizontal tangent lines at (0, -9) and (4,-41).a=b=c=d=
- Q Find dy/dx by implicit differentiation.9x3 + xy-xy = 8
- Q Find the equation of the line that is tangent to the graph of the given function at the specified point.y = x^4 - 5x^3 + 4x + 4; (1, 4)y=-11x...
- Q Let f(x) = x + 64/x+6(a) Find the domain of f. (Enter your answer using interval notation.)The domain of f is(b) Find the derivative of the function f.(c) Find the...
- Q Let f(x) =x-6/x + 1(a) Find the derivative of the function f. (Simplify your answer completely.)f'(x) =(Express your answer as a single fraction.)(b) State the domain off and the domain...
- Q Use the Product and Quotient Rules to find the derivative of each function.1. f(x) = 3x cos x2. y = 7?x sinx3. y secx tan x4. f(x) = x+2 /...
- Q Businesses can buy multiple licenses for a data compression software at a total cost of approximately C(x) = 48x^2/3 dollars for x licenses. Find the derivative of this cost function...
- Q Use the alternative form of the derivative to find the derivative at x = c (if it exists). (If the derivative does not exist at c, enter UNDEFINED.)f(x) = x...
- Q For the function f(x) = 3x + 2x - 6, evaluate and fully simplify each of the following.f(x + h) =f(x+h)-f(x)/h=
- Q Give the average rate of change for f(x) = 2x - 12x - 21 between x = 2 and x = 5.The average rate of change is
- Q Find the difference quotient f(x+h)-f(x)/h for the function given below.f(x) = 9x - 1Give a simplified expression involving x and h, if necessary. For example, if you found that the...
- Q A ball is thrown from a height of 156 feet with an initial downward velocity of 8 ft/s. The ball's height h (in feet) after t seconds is given by...
- Q Find the indefinite integral by making a change of variables.We are given the following indefinite integral and are told to make a change of variables to find the integral.It is...
- Q Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. (Enter your answers using interval notation. If an answer cannot be expressed as an interval,...
- Q Find the general solution of the differential equation and check the result by differentiation.dy/dt = 16t^7
- Q Fill in the missing interpretations in the table using the example as a guide.A. The average rate of temperature change between to and toh hours after midnight. The instantaneous rate...
- Q Suppose u and v are functions of x that are differentiable at x = 0 and that u(0) = 1, u'(0)=9, v(0) = -3, and v'(0) = 8. Find the...
- Q Find the derivative of the function by the limit process.g(t) = t + 3t
- Q Find the two x-intercepts of the function f and show that f'(x) = 0 at some point between the two x-intercepts.f(x) = x(x - 9)(x, y) =(x, y) =(smaller x-value)X...
- Q A person standing close to the edge on top of a 16-foot building throws a ball vertically upward. The quadraticfunction h(t) = - 16t + 60t + 16 models the...
- Q Find f'(x) and f'(c).Functionf(x) = (x + 2x)(4x4 + 4x ? 3)f'(x) =f'(c) =Value of cc = 0
- Q Consider the function on the interval (0, 2?).(a) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)(b) Apply the First Derivative...
- Q Find the second derivative of the function.f(x) =cscxf"(x)=
- Q Consider the function on the interval (0, 2?).f(x) = x/2 + cos x(a) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval...
- Q Find the derivative of the function.f(t) = arccsc(-2t) f'(t) =2/ t 4t4-1
- Q Find any critical numbers of the function. (Enter your answers as a comma-separated list.)g(x) = x8 - 4x6
- Q Find the derivative of the function.y = arctan x/2 - 1/2(x + 4)
- Q Verify that f has an inverse function. Then use the function f and the given real number a to find (f )'(a). (Hint: See Example 1. If an answer does...
- Q Use logarithmic differentiation to find dy/dx.y = (x + 1)(x - 6) / (x - 1)(x + 6)
- Q Find dy/dx by implicit differentiation and evaluate the derivative at the given point.x + y = 16xy - 3, (8,5)
- Q The equation of the line tangent to the graph of f(x) = 5-x at x = 1 isy = -2x + 1.y = -2x + 3.y=-2(In 5)x+ 1.y = -2(In...
- Q Find the derivative of the function.h(x) = x^2 arctan(5x)
- Q Find dy/dx at the given point for the equation.x = y - 9y + 2, (-6, 1)Step 1The given function is x = y3 - 9y + 2. Differentiate the...
- Q Find the two x-intercepts of the function f and show that f '(x) = 0 at some point between the two x-intercepts.f(x)=x-3x - 18(x, y) = () (smaller x-value)(x, y)...
- Q Find the derivative of the function.f(t) = arccsc(-2t)
- Q Find the critical numbers of the function.f(x) = 5x - 4xStep 1To find the critical numbers of f, you must find all x-values for which f'(x) or f'(x)
- Q The minimum and maximum of a function on an interval are the extreme values, or extrema (the singularform of extrema is extremum), of the function on the interval. In the...
- Q To find where f'(x) = 2(5x - 2) is equal to 0, we check where 5x 2 is equal to 0.5x ? 2 = 0 x =Because f'(x) exists for...
- Q Find dy/dx by implicit differentiation.x4 + 5 = 15With implicit differentiation, we differentiate all the terms in our equation with respect to x. Keep in mind we are implying that...
- Q Find the derivative of the function.h(s) = -2?3s + 2
- Q Find an equation of the tangent line to the graph of the equation at the given point.(-2, 1)x + x arctan y = y - 1,Step 1Use implicit differentiation to...
- Q To find the derivative of the left side of the equation, rewrite the square root using exponents and use thePower Rule. Use the Sum Rule and the Constant Multiple Rule...
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