Problem: After being built, a car must be painted. The revenue, R, in dollars, when x cars are painted can be modelled by the function R(x) = 1000x – 0.01x2. a) Determine the average rate of change of revenue when painting 20 to 50 cars.  b) Estimate the instantaneous rate of change of revenue after painting 50 cars.  c) Interpret the results found in parts a) and b)

Q: y = 3x?+1 ; (0,1)

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Q: Find all points on the graph of f(x) = x +a where the tangent line is parallel to the line y = 28x.…

A: Consider the function f(x) = x3 + x Slope of the tangent to f(x) is the derivative of f(x). f'(x) =…

Q: The slope of the tangent line to the graph of the function y = 2/x – 6 at (36, 6) is 6 -0.50 0.17…

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Q: The slope to the tangent line of a curve is given by f(x) = x2 -9x + 12. If the point (0, 9) is on…

A: Given that slope equation and we find the equation of curve. I.e.  df(x)/dx = x2 - 9x + 12. And also…

Q: 1. FIND THE SLOPE OF THE TANGENT LINE TO THE GRAPH OF THE FUNCTION f(x) = 5 – x² AT (3,–4).

A: f(x)=5-x2 To find the slope of tangent, we can use two alternatives (1) Direct derivative (2)…

Q: Show that an equation of the tangent line to the graph of =D2 at the point (a, b) is given by 2. b…

A: It is given that, Now take the derivative of the given function with respect to 'x'. Therefore,…

Q: Find the slope of the tangent line to f(x) = e^[3 (x^2) − 4x] at the point (0,1).

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Q: For which value(s) of x does f(x)=−x^3/3−2x^2+3x−8 have a tangent line of slope 7?

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Q: 9) Find the equation of the tangent line to the function f(x)=-3x² - 2x at the point where x=1. Give…

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Q: Find the equation of the line that is tangent to the graph of the function f(x) = ln(8x + 6) at the…

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Q: Compute f ' (x) and find an equation of the tangent line to the graph at x = a. f(x) = 3√x,  a= 8

A: Derivatives can be defined as rate of change of function with respect to an independent variable. It…

Q: Find the equation of the tangent line to the graph of the function f (x) = (x² + 8) (x – 3) at the…

A: The gradient of the tangent at any particular point is given by :     f(x) = (x2 + 8) (x - 3)…

Q: 60. Find an equation of the tangent line to the graph of f(x) a) at (1, 1); b) at (3, 5); c) at…

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Q: #36 Find the equation of the tangent line to the graph of X²(x²+y²) = y? at

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Q: 2) Find the equation of the tangent line to the graph of x? + (y – x)3 = 9 at the point x = 1.

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Q: 2. Determine the equation of the line perpendicular to the tangent line to the function f(x) = e' at…

A: We will find the equation of the line.

Q: What represents the slope of the tangent to f(x)= √x+3 at x=1

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Q: Find an equation of the line tangent to f(x) at the point |3D 1+e

A: Find slope at given point and then use formula  to find tangent line

Q: 34. Find an equation of the line tangent to the graph of f(x) = (2x – 1)(x + 4) at (1, 5).

A: The objective of the question is find the equation of tangent at the point (1,5).

Q: Find the slope of the tangent to the graph of fAx) = x + 3x + x at its point of inflection. O a) -3…

A: SOLUTION-

Q: 22. Find an equation of the tangent line to the graph of f(x) = 4 - x² at (a) (-1,3); (b) (0, 4);…

A: Given: f(x)=4-x2 To Find: the equation of the tangent line to the graph at a) (-1, 3) b) (0, 4)…

Q: Find the equation of the tangent line to the graph of f (x) = x² – 3 at the point (3, 6). Answer y =

A: Find the equation  of the tangent  line to  the graph  offx=x2-3   at the point  3,6Answer:We  have…

Q: the

A: The given function is f(x)=x+1x-6at the point 0,-16 To find the equation of the tangent line the…

Q: 10. (a) Find an equation for the line tangent to f(x) = x³ + 4x² at the point x = 1.

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Q: Solve the problem. Find where the tangent to the curve f(x) = V+1 that has slope . O x = 3 O x = 1 O…

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Q: A) Find an equation of the line tangent to the function of x2 + (y – x)² = 9 at x = 1.

A: Here we find equation of tangent on given curve.

Q: 2. Given the function, find the slope of the line tangent to the curve: f(x) = v5x + 2 at the point…

A: We have to find the slope of the line tangent to the curve: f(x)=5x+2 at the point P(1,2) for the…

Q: Find an equation of the tangent line to the graph of the function f(x) = esec(x) at the point (*,…

A: We have to find the equation of tangent line to the graph of function fx=esecx at the point π4, e2.…

Q: Let f(x) = 6 - 3x + 5x2. (a) Find f'(a) (b) What is the instantaneous rate of change of f when x =…

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Q: 19. Find an equation of the tangent line to the graph of f(x) = 2/x at (a) (1, 2); (b) (–1, –2); (c)…

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Q: Find all values of x where the slope of the tangent line to f(x)=4x^2−3x+2 is 0. x = ____

A: The detailed solution is given below Thank you

Q: Compute f'(x) and find an equation of the tangent line to the graph at x = a. f (x) = 5x − 32√x, a =…

A: For a function f(x), f'(x) denotes its first derivative. Value of first derivative at a point gives…

Q: Find an equation for the line tangent to the graph of…

A: Given function is fx=-2xex To find: Th equation of tangent line at point a,fa for a=2 Solution:…

Q: the line tangent to f(x)=x' 3.

A: we have to find the equation of line tangent to  f(x)=x3-x2-5      , at the point (2,-1)

Q: The slope of the tangent line to the graph of f(x)%3D at a =-1 is: x2 - Select one: 1. a. O b. O c.0

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Q: Find an equation of the line tangent to the graph of f at the given point.

A: Solution:Given          f(x)=sec-1(ex);              (ln 2, π/3)To find the equation of the line…

Q: (4) Find the equation of the line tangent to f(x) = 12√ - 2² at the point (4,8).

A:

Q: A line with slope m passes through the origin and is tangent to the graph of y = ln x. What is the…

A: Let the point of contact is (h,ln(h)) We find the slope of the tangent using dy/dx

Q: Suppose f (x) = (x 2 − 2x) (3x + 2).What is the equation of the line normal to f(i.e., the line…

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Q: a) Find the points at which the tangent line to the graph of function f(x)=2x²-24x+6 has slope equal…

A: Let y=f(x) be any function, then    Slpoe at any point, m = df(x)dx

Q: Use Definition 1 or 2 to find the instantaneous rate of change of the function at the given value.…

A: The instantaneous rate of the function N(x)=1x, x=8

Q: A. Find the slope of the line tangent to the given curve at the indicated point. 1. f(x) = 3x + 10…

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Q: Find the equations of the tangent line and normal lines to the graph of the function f(x)=6sinx at…

A: y = f(x) = 6sinx We have to find the equations of the tangent line and normal lines to the function…

Q: Find the slope of the tangent to Ax) = x²e* at the point (1, e). a) e b) 1 c) 3e d) 2 + e

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Q: 2. Find the equation of the tangent line to the graph of f(x) = 3x?e-1 - 1, at r = 1.

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Q: Solve the problem. Find where all tangents to the curve f(x) = that have slope -1. x + O x = 0 O x =…

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Q: Find the equation of the tangent line to f(x)=4x+x^−1/2+1 at the point (4,35/2)

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Q: Find an equation for the line tangent to the graph of: f(x)= 2xex at the point (a,f(a)) for a=2.

A: Given curve Now, we find f'(x) We know that slope of tangent line to the curve y=f(x) at point x=a…

Q: Find an equation of the tangent line to the graph of the function h(x) = −xe2−x, at the given point…

A: Slope of the tangent line is the derivative of the given curve at the given point. So…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.