Given that  f  is a function of  x, which of the following statements is false? a) When  f  is not changing, df/dx is zero. b) When  f  is decreasing, df/dx is negative. c) When  f  is decreasing, df/dx is positive. d) df/dx denotes the rate of change in  f  with respect to  x .

Q: Use the limit definition to compute f '(a) and find an equation of the tangent line. f(t) = t-3, a =…

A: Given: ft=t-3  ……1 a=1  ……2   To find: f'a and equation of tangent line

Q: Assume that is differentiable on a ≤ x ≤ b and that f (b) < f (a). Show that f' is negative at some…

A:

Q: Consider the following function. f(x) = %3D Find its average rate of change over the interval [9,…

A: We have to find ∆y∆x , f'(9) and f'(16)  Given f(x)=1x

Q: If f(x) = In(tanh(2æ)), then f'(x) is %3D Select one: O a. coth(2a)sech (2r) O b. 2 coth(2a)sech…

A: Given: f(x) = ln(tanh(2x))…

Q: If f' is continuous on [a, b], show that 2 f(x)f'(x) dx = (F(b))? – (F(a)² %3D -

A:

Q: Let A E R"Xn, let b e R", and let n n f (x) = xT Ar + 6Tx = Aikr;xk +> b;x;. i,k=1 i=1 Show that f :…

A: Definition: A function f:ℝn→ℝm is differentiable at x0 with derivative M if M:ℝn→ℝm is linear and…

Q: Consider the function f(x) = s x (1 – x) with x E [0, 1] . Is f contractive on its domain? Make sure…

A: Contractive map

Q: a) At how many points is f(z) discontinuous? b) At how many pints is f(z) not differentiable?

A: This is a calculus problem and I solved it in the next step.

Q: Find a value of x such that the following relation holds true. dt dt X =

A:

Q: Let f(x) ∈ R[x]. If f(a) = 0 and f '(a) = 0 [f '(a) is the derivative off(x) at a], show that (x -…

A:

Q: Where is the function f(r, y) = x/y continuous?

A:

Q: Use the limit definition to compute f '(a) and find an equation of the tangent line.

A:

Q: Let f(x) = Vx and L(x) be the tangent line approximation of f(x) at x =9. Which of the following…

A: Given that       f(x) = √x  L(x) be the tangent line approximation of f(x) at x=9 . Here we have to…

Q: Find linearization L(x) of f(x) = tanh=' at a= 0 and use it to estimate f(0.01).

A: Topic:- linearisation

Q: Let F(x) be the function defined as follows: F(x)= (1+ In(t))*dt What is the value of F'(e) ?

A:

Q: Is the function F (x) = L", e“ dt concave (concave down) or convex (concave up) for x<0?

A: To solve the following problem

Q: Answer the following True or False: Let f(x) be a differentiable function, such that f(– 1) = - 5…

A:

Q: Suppose f(x) is continuous and increasing on [a, b], and concave up on (a, b). Is Lo (the Left-Hand…

A:

Q: Consider the function F(x) = | s(t) dt , where a is a constant and f is continuous everywhere. By…

A: Find we have to use definition of derivative.

Q: Provide an appropriate response. Why is dy not generally equal to Ay? Because Ay is the slope of…

A:

Q: Let f(x) E R[x]. If f(a) = 0 and f'(a) = 0 [f'(a) is the derivative of f(x) at a], show that (x –…

A: Given: f(a) = 0, f'(a) = 0 [f'(a) is derivative of f(x) at a] To Show: (x-a)2 divides f(x) Concept…

Q: If f:R to {0,1} and I is an interval on which f has both values , the f has a discontinuity in I.…

A: Given that f has both 0 and 1 in its range.  This means there will be some x and y values for which…

Q: Show that the subtangent of f (x) = ex is everywhere equal to 1.

A: Formula to find the subtangent of the function helps to find the required subtangent. Derivative of…

Q: Let f(x) Vr +2 and use the Linear Approximation to this function at a = 2 with Az = 0.5 to estimate…

A: Given that: f(x)=x+2a=2 and ∆x=0.5

Q: Suppose f: a,b] → R is continuous and faf=/ for all xie [a,bl Show that f(x)= 0 ft rEla,b).

A:

Q: Let f be a differentiable function of one variable, and let w = f(p), where p = (x² + y² + z?)\/2.…

A:

Q: Use the limit definition to compute f 1 (a) and find an equation of the tangent line. f (t) = 2…

A:

Q: The height of a ball t seconds after it is thrown upward from a height of 7 feet and with an initial…

A: solution :- The height of a ball t seconds after it isthrown upward from a height of 7 feet andwith…

Q: is continuous and integral 3 to 1 f(x)dx =8, show that the graph of f intersects the line y=4 at…

A: Given that f is continuous and f13f(x)dx=8 To prove line y=4 intersects the graph of f altleast can…

Q: 2. For which values of a is f(x) differentiable on R?

A:

Q: The two point forward difference formula to approximate f'(xo) with h<0 is f(xo +h) – f(xo) h

A: f'(a)=difference in the y-valuesdifference in the x-values      = f(a+h)-f(a)h This is called a…

Q: Suppose f(r) = (In z)e. What is f'(x)?

A: d/dx (f(x)g(x))= f(x)g'(x)+g(x)f'(x)

Q: Let f(x) E R[x]. Suppose that f(a) = 0 but f' (a) + 0, where f' (x) is the derivative of f(x). Show…

A:

Q: Establish the interval where the function h ( x) *x° + 1 is concave down (CD), and find its point of…

A:

Q: Prove that is a function f is harmonic =0, then - dx -Ldy =0.

A: Given function f is harmonic such that ∂2f∂x2+∂2f∂y2=0

Q: 3. Find the lincar approximation of the function g(x) = V1+x at a = 0 and use it to approximate the…

A: Use taylor form

Q: Prove that the function z is continuo:as everywhere but nowhere differentiable except at origin.

A: The given problem is solved below in detail.

Q: Use the limit definition to compute f '(a) and find an equation of the tangent line.

A:

Q: x+ (х,у) # (0,0) f(x,у) — 1-/x² +y² +1 k (х, у)%3 (0,0)

A:

Q: Use the limit definition to compute f '(a) and find an equation of the tangent line.

A:

Q: Determine the intervals on which the function F(r) = te dt is concave up or down.

A: Consider the given integral as, Fx=∫4x+5t7e-3tdt As we know that if F"x&gt;0 the function will be…

Q: a). Find the linear approximation to f(x) = VI at r = 64, (b). Use it to approximate V65. c).…

A:

Q: Is the given function f (x, y, z) = xª – 6xy? + 723 continuous at the origin? Yes No

A: If a function f and another function g is continuous at arbitrary point. Then there sum, difference…

Q: (1) Where on the line r = ay does the function f(r, y) = (x – a)² + (y – 2)² attain its smallest…

A: We need to check on the line x=ay.....................(1) if there exist a minimum for the function…

Q: Suppose that f:I → R and g: I → R are differentiable at c E I. Then, a) If k E R, then the function…

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Q: Answer the following True or False: Let f(x) be a differentiable function, such that f( – 5) = with…

A:

Q: Suppose w is a function of x, y, and z, which are each functions of t. Explain how to find dw/dt.

A: Given data:"w" is a function of (x,y,z). Therefore, w(x,y,z). And the function x, y and z are the…

Q: d. Use the limit definition of the derivative, f (x) = lim f (x+h)-f(x) 1 +h)=j, to prove that VT =…

A:

Q: Assume that ƒ is differentiable on a&lt;=x&lt;=b and that ƒ(b)&lt;ƒ(a). Show that ƒ′ is negative at…

A: Given,

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.