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Nobuhle Tavonesa

Nobuhle T.

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A bug is moving along the right side of the parabola $y=x^{2}$ at a rate such that its distance from the origin is increasing at $1 \mathrm{cm} / \mathrm{min.}$ a. At what rate is the $x$ -coordinate of the bug increasing at the point (2,4)$?$ b. Use the equation $y=x^{2}$ to find an equation relating $\frac{d y}{d t}$ to $\frac{d x}{d t}$ c. At what rate is the $y$ -coordinate of the bug increasing at the point (2,4)$?$

A bug is moving along the right side of the parabola $y=x^{2}$ at a rate such that its distance from the origin is increasing at $1 \mathrm{cm} / \mathrm{min.}$ a. At what rate is the $x$ -coordinate of the bug increasing at the point (2,4)$?$ b. Use the equation $y=x^{2}$ to find an equation relating $\frac{d y}{d t}$ to $\frac{d x}{d t}$ c. At what rate is the $y$ -coordinate of the bug increasing at the point (2,4)$?$

Calculus: Early Transcendentals

Derivatives

Related Rates

Use Version 2 of the Chain Rule to calculate the derivatives of the following composite functions. $$y=\sqrt{10 x+1}$$

Calculus Early Transcendentals

Derivatives

The Chain Rule
Use Version 1 of the Chain Rule to calculate $\frac{d y}{d x}$. $$y=\sqrt{x^{2}+1}$$

Use Version 1 of the Chain Rule to calculate $\frac{d y}{d x}$. $$y=\sqrt{x^{2}+1}$$

Calculus for Scientists and Engineers: Early Transcendental

Derivatives

The Chain Rule

Differentiate the function. $ F(r) = \frac{5}{r^3} $

Differentiate the function. $ F(r) = \frac{5}{r^3} $

Calculus: Early Transcendentals

Differentiation Rules

Derivatives of Polynomials and…

Questions asked

INSTANT ANSWER

2. Let \( g(x)=x^{2} e^{-x} \). (i) Determine the intervals of increase asd decrease for \( g(x) \). (ii) There are two eritical points of \( g(x) \). Use the First Derivative Test (that is, use part (i) to classify th critical points as relative max/min or neither. (iii) Determine the intervals of concave up and down for \( g(x) \). Give the inflection points of \( g(x) \).

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INSTANT ANSWER

a) Set up the formula for Newton's Method. Write your formuls in the form: \[ \begin{array}{l} x_{n+1}=x_{n+1}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} \end{array} \] (ii) First appraximate the positive oolution. Using the starting goess \( x_{4}=1 \), calculate \( x_{1} x_{2}, x_{2} \ldots \ldots \) until the result agrees to 6 decimal places. (iii) Now approximate the negative solution. Choose a suitable initial cuess for \( x_{0} \), then calculate \( x_{1}, x_{2}, x_{3}, \ldots \) until the result agrees to 6 docimal places.

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ANSWERED

Zhumagali Shomanov verified

Numerade educator

Assume ( x ) and ( y ) are functions of ( t ). Evaluate ( frac{d y}{d t} ) for ( 3 x e^{y}=9-ln 729+6 ln x ), with the conditions ( frac{d x}{d t}=12, x=3, y=0 ) [ frac{mathrm{dy}}{mathrm{dt}}=square ] (Type an exact answer in simplified form.)

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ANSWERED

Bcrypt_Sha256$$2B$12$We1Wwocamog01O5I.V2Tkouxdh4Ofnmgpwkor7Leaonfpu0Ubfpua Bcrypt_Sha256$$2B$12$We1Wwocamog01O5I.V2Tkokttmmj7Lscvwvlptp4Rlhbswcdg9.Wy verified

Numerade educator

(b) [2 pts] The annual sales (in thousands of dollars) of a certain product at time ( t ) (years after year 2000 ) is given by: [ S(t)=frac{20}{1+30 e^{-t / 5}} ] (i) Find the rate of change of ( S(t) ) (in dollars per year, rounded to the nearest dollar) in year 2020. (ii) In the long run (as ( t ) becomes large), what value does ( S(t) ) approach?

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ANSWERED

Vincenzo Zaccaro verified

Numerade educator

2. (a) [3 pts] ( f(x) ) and ( g(x) ) are given by the graphs below: Let ( h(x)=f(g(x)) ) and ( k(x)=f(x) g(x) ). Use the graphs to evaluate the following quantities: (i) ( f(1)= ) (ii) ( f^{prime}(1)= ) (iii) ( f(4)= ) (iv) ( f^{prime}(4)= ) (v) ( g(4)= ) (vi) ( g^{prime}(4)= ) (v) ( h^{prime}(4)= ) (vi) ( k^{prime}(4)= )

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ANSWERED

Zhumagali Shomanov verified

Numerade educator

(b) ( [1+2 mathrm{pts}] ) Find the following derivatives. Show your work and clearly state the rules that are applied. Simplify your answer whenever reasonable. (i) Find ( frac{d}{d x} e^{3 x} sin left(x^{2} ight) )

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ANSWERED

Zhumagali Shomanov verified

Numerade educator

(ii) Find ( frac{d y}{d x} ), where ( y=ln left(sqrt{x^{2}+1} ight) ). (Hint: View ( y(x) ) as a chain of functions ( x ightarrow u ightarrow v ightarrow y ).)

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INSTANT ANSWER

1. (a) [3 pts] An object is thrown upward from the ground with the given height function ( ( t ) is in seconds, ( h(t) ) is in metres): [ h(t)=26 t-5 t^{2} ] Find the average speed over the following intervals of time (you may use a calculator): (i) ( [1,1.5] ) (ii) ( [1,1.1] ) (iii) ( [1,1.01] ) (iv) ( [1,1.001] ) (v) ( [1,1.0001] ) Based on the above calculations, what does the instantaneous speed appear to be at ( t=1 ) second?

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ANSWERED

Zhumagali Shomanov verified

Numerade educator

Use the table below to find frac{d}{dx}[2x - 5f(x)]Big|_{x = 5}. x 1 2 3 4 5 f'(x) 4 2 5 1 3 g'(x) 2 1 4 3 5 frac{d}{dx}[2x - 5f(x)]Big|_{x = 5} = square (Type an integer or a decimal.)

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ANSWERED

Keondre Parker verified

Numerade educator

The total cost (in hundreds of dollars) to produce ( x ) units of a product is ( C(x)=frac{9 x-5}{8 x+3} ). Find the average cost for each of the following production levels. a. 5 units b. ( x ) units c. Find the marginal average cost function. The average cost for 5 units is ( $ square ) per unit. (Round to the nearest hundredth as needed.)

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