Question

The function f(x) = sin (5x^2 ) does not have an antiderivative in terms of the commonly known functions.

Use the Fundamental Theorem of Calculus to find a function F(x) satisfying F'(x) = sin (5x2 )and F(4) = 5 .

Answer #1

1. If f(x) = ∫10/x t^3 dt then: f′(x)= ? and f′(6)= ?
2. If f(x)=∫x^2/1 t^3dt t then f′(x)= ?
3. If f(x)=∫x3/−4 sqrt(t^2+2)dt then f′(x)= ?
4. Use part I of the Fundamental Theorem of Calculus to find the
derivative of h(x)=∫sin(x)/−2 (cos(t^3)+t)dt. what is h′(x)= ?
5. Find the derivative of the following function:
F(x)=∫1/sqrt(x) s^2/ (1+ 5s^4) ds using the appropriate form of the
Fundamental Theorem of Calculus.
F′(x)= ?
6. Find the definitive integral: ∫8/5...

Calculus, Taylor series Consider the function f(x) = sin(x) x .
1. Compute limx→0 f(x) using l’Hˆopital’s rule. 2. Use Taylor’s
remainder theorem to get the same result: (a) Write down P1(x), the
first-order Taylor polynomial for sin(x) centered at a = 0. (b)
Write down an upper bound on the absolute value of the remainder
R1(x) = sin(x) − P1(x), using your knowledge about the derivatives
of sin(x). (c) Express f(x) as f(x) = P1(x) x + R1(x) x...

1. Let f(x)=−x^2+13x+4
a.Find the derivative f '(x)
b. Find f '(−3)
2. Let f(x)=2x^2−4x+7/5x^2+5x−9, evaluate f '(x) at x=3 rounded
to 2 decimal places.
f '(3)=
3. Let f(x)=(x^3+4x+2)(160−5x) find f ′(x).
f '(x)=
4. Find the derivative of the function f(x)=√x−5/x^4
f '(x)=
5. Find the derivative of the function f(x)=2x−5/3x−3
f '(x)=
6. Find the derivative of the function
g(x)=(x^4−5x^2+5x+4)(x^3−4x^2−1). You do not have to simplify your
answer.
g '(x)=
7. Let f(x)=(−x^2+x+3)^5
a. Find the derivative....

1) Find the antiderivative if f′(x)=x^6−2x^−2+5 and f(1)=0
2)Find the position function if the velocity is v(t)=4sin(4t)
and s(0)=0

(a) Find the most general antiderivative of the function f(x) =
−x^ −1 + 5√ x / x 2 −=4 csc^2 x
(b) A particle is moving with the given data, where a(t) is
acceleration, v(t) is velocity and s(t) is position. Find the
position function s(t) of the particle. a(t) = 12t^ 2 − 4, v(0) =
3, s(0) = −1

Verify that the function f(x) = 5x 3 − 2x − 4 satisfies the
hypotheses of the Mean Value Theorem on the interval [−2, −1]. Then
find all numbers c that satisfy the conclusion of the Mean Value
Theorem.

(a) Is the vector field F = <e^(−x) cos y, e^(−x) sin y>
conservative?
(b) If so, find the associated potential function φ.
(c) Evaluate Integral C F*dr, where C is the straight line path
from (0, 0) to (2π, 2π).
(d) Write the expression for the line integral as a single
integral without using the fundamental theorem of calculus.

Find derivatives (Please show work!)
1. f(x)=ln(5x^3)
2. f(x)=(ln^3)x
3. f(x)=e^(5x2+2)
4. f(x)=xe^2x
5. f(x)=xlnx

Use logarithmic differentiation to differentiate the following
function.
f(x)=(x+7)5(5x-2)4
f'(x)=___________

1a.) Find the linearization of the function f(x) = (sin x+1)^2
at a = 0.
1b.) Differentiate the two functions below.
f(x) = ln(e^x - sin x) ; g(x) = e^-x^2

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