For the functions f(t) = u(t) and g(t) = u(t) (where u(t) is the Heaviside function), defined on 0

Q: the function f is not differentiable at x=0.

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: Sketch the graph of a function f such that f (x) ≥ 0 on [0, 1] and f (x) ≤ 0 on [1, 2], whose…

A: Consider the function Then and The graph of the function is shown below,

Q: Consider the functions f(t) = e' and g(t) -81 defined on 0 st<o. %3D = e (a) (f* g)(t) can be…

A: Consider the functions, f(t) = et and g(t) = e-8t

Q: QIFind the derivative of the function f(z, k) = z* – 16k + tanh(z) – 4 csc(k) in terms of z and k ?

A: Dear student  Here is the solution of your query

Q: Let f be a function on [0, 7] whose second order derivative is continuous and f(r) = 1. Suppose that…

A: We have to solve given problem:

Q: Suppose that f'(x) > M for all x E [a, b]. Prove that f(b) > f(a) + M(b

A: Since  f'(x)≥M therefore the function is differentiable and hence continuous let f is continuous on…

Q: If f is an entire functions then h(z) = e² f(1/2²) is analytic on C\ {0}, and its derivative h'(2) =…

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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: Calculate Rs for f(x) = 6 - x over ver [3,5]. R8 IE

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Q: [59] Evaluate F.dr where F(x.y)3y'i+(x-4)j and C is the portion of y=(x-1) from .x-0 to r-3.

A: Please see the explanation below

Q: Let f(x): = [2x¹, -ㅠ < x < 0 (5sin(x), 0 ≤ x < ㅠ 음ㅠㅠ 15 - 10 83 – 10 ㅇ ㅠㅠ 25 +10 Evaluate the…

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Q: Which of the following is the domain of f(x, y) = ln(9-x² - y²)? Ox² + y² ≤9 Ox² + y² ≥9 00< x² + y²…

A: To find the domain of the following function.

Q: Consider the function f(x) = x(x² + 3)". (a) Use the Fundamental Theorem of Calculus to evaluate |…

A: (a) Consider the given function as fx=xx2+37

Q: Suppose that f: R² → R² is given by f(x, y) = (x³ - 2xy², x+y). Does f has a differentiable local…

A: Inverse mapping theorem.

Q: et f(x) = 6x? on [(1, 3.5). a) Find L(f, P) and U(f, P) when P 3.5 p) Use calculus to evaluate 6x…

A: In this question, the concept of Riemann Integral is applied. Riemann Integral   To estimate…

Q: Prove that for any function f : X → Y and for any subsets Y₁ and Y₂ of Y, ƒ¯¹(Y₁ NY₂) = ƒ¯¹(Y₁) Ñ…

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Q: Let f(t) be the velocity in meters/second of a car at time t in seconds. Give an integral for the…

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Q: Let f(x) — sin(nx) n³ n=1 Show that f is differentiable on R and that f' is continuous on R.

A: Within its interval of convergence, the derivative of a power series is the sum of derivatives of…

Q: Let z= x+iy and determine where the function f(z) = Log(z2 + 1) is holomorphic. What is its…

A: The derivative of f(z) is given by the chain…

Q: Use the MVT to prove that if f' (x) ≤ 2 for x &gt; 0 and f (0) = 4, then f (x) ≤ 2x + 4 for all x ≥…

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Q: The graph of the function f consists of the three line segments joining the points (0, 0), (2, −2),…

A: (a) Plot the points on the graph and join the line segments to construct the graph of f(x).

Q: Assume that f and g are differentiable functions such that f (0) > g(0) and that for all x E R,…

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Q: Use a summation to find the integral from 0 to 4 of f(x) 2x + 1.  Then check it using the power rule…

A: Integration and derivative

Q: Suppose that f(x) is continuous on [0, 9] and F(x) is an anti-derivative of f. If F(0)=9 and F(9)=3,…

A: We have to find

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.