Calculus Answers

Evaluate LaTeX: \int_cF.dr\:\: where LaTeX: F\left(x,y,z\right)=xzi-yzkF(x,y,z)=xzi−yzk and c is the line segment from (3,0,1) to (-1,2,0)

Find all The points on the graph of (x^2+y^2+y)^2=x^2+y^2 that have horizontal tangent line

Find all the points on the graph of (x^2 + y^2)^2 = x^2−y^2 that have horizontal tangent line.

Find the slope of the curve at the given point P and an equation of the tangent line at P.

y = -3 - x³, (1, -4)

Let y be defined implicitly by the equation ln(3y)=5xy Use implicit differentiation to find the first derivative of y with respect to x

a)dy/dx=

Use implicit differentiation to find the second derivative of y with respect to x

b)d2y/dx2= 

c)Find the point on the curve where d2y/dx2=0

2- The longitudinal extension of metal bar in direction of an applied force is

given by

y = LKeF×1×10−3

Where y is the longitudinal extension in m, L is the length of the bar in m which

is L= 0.15 (m), K is a constant depends on the material and is K= 1, and F is applied

force in N.

a) Find the work done if the force increases from 100 N to 500 N using:

i) An analytical integration technique

ii) A numerical integration technique (n=8 intervals)

[Note: the work done is given by the area under the curve]

b) Using a computer spreadsheet and recalculates step (ii) by increasing

the number of intervals to n=10 and compare your obtained results

with (i) and (ii)

c) Using Simpson’s rule to find the work done (n=8).

d) Analyse whether the size of numerical steps has effect on the obtained

result and explain why.

Evaluate the following integrals:

1. ∫ 𝑡 𝑐𝑜𝑠2𝑡 𝑑𝑡
2. ∫ 𝑥^3 𝑙𝑛𝑥 𝑑𝑥
3. ∫ 𝑥^3 𝑒^3𝑥 𝑑𝑥
4. ∫ 𝑠𝑖𝑛(𝑙𝑛𝑥) 𝑑𝑥
5. ∫𝑠𝑖𝑛^−1 𝑥 𝑑𝑥
6. ∫ 𝑦^3 𝑒^𝑦^2 𝑑𝑦

xy^2 +5 = 9x−3y^2

1) The quantity of a substance can be modeled by the function k(t) that satisfies the differential equation  dk/ dt =-1/90 (k-450). One point on this function is k(2) = 900. Based on this model, use a linear approximation to the graph of k(t) at t = 2 to estimate the quantity of the substance at t = 2.1.

2) Consider the differential equation dy/dx =x^2y^2with a particular solution y=f(x) having an initial condition y(−3) = 1 . Use the equation of the line tangent to the graph of f at the point (−3, 1) in order to approximate the value of f(−2.8) .

3) Given the differential equation dy/dx=-x/y^2 find the particular solution, y=f(x) , with the initial condition f(−6) = −3 .

4)  What is the particular solution to the differential equation dy/dx =3 cos(x)y with the initial condition y(pi/2)=-2?

Find the equation of the tangent line and normal line to the curve -2e^-x + e^y = 3e^x-y at point (0,0)