Using limit to find the equation of the tangent line to the curve y = f(t)=2t^3+3t at the point Q(1,5)

y=x^5 - 5x

Use the "Guidelines for sketching a curve A-H"

A.) Domain
B.) Intercepts
C.) Symmetry
D.) Asymptotes
E.) Intervals of increase or decrease
F.) Local Maximum and Minimum Values
G.) Concavity and Points of Inflection
H.) Sketch the Curve

1. Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = x³ + y³ − 3x² − 6y² − 9x

2. Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = 3x³ − 9x + 9xy²

^3. Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f (x, y) = xy − 3x − 3y − x² − y²

3. Jose deposits $150 at the end of each
month into an account that pays 6%
interest compounded monthly . After 8
years of deposits, what is the account
balance?

Prove the following using the method suggested:
(a) Prove the following either by direct proof or by contraposition:
Let a ∈ Z, if a ≡ 3 (mod 5) and b ≡ 2 (mod 5), then ab ≡ 1 (mod 5).

(b) Prove the following by contradiction:
Suppose a, b ∈ Z. If a² + b²
is odd, then (2|a) ⊕ (2|b), where ⊕ is the exclusive disjuntion,
i.e. p ⊕ q = (p ∨ q) ∧ ¬(p ∧ q).

(d) Prove the following by cases: For all n ∈ Z, n
2 + 3(n + 1) is odd.

(e) Prove the following by induction:
For n ≥ 1,
1 × 2 + 2 × 3 + 3 × 4 + · · · + n(n + 1) = n/3
(n + 1)(n + 2)

C. Clearly indicate the plate number and indexing movements to be made to divide the periphery of a work piece into 29 divisions using compound Indexing? The following indexing plates are available. Plate No 1: 15,16,17,18,19,20.: Plate No 1: 21,23,27,29,31,33. Plate No 1: 37,39,41,43,45,49

Evaluate the integral by making an appropriate change of variables.

where R is the trapezoidal region with vertices (6, 0), (10, 0), (0, 10), and (0, 6)

Differentiate each. give reasonably simplified answers. Box ANSERS.

(a)=ln[(x^9*(5x+1)^4*(11x+2))/((8x+9)(3x^5-2x+1))]

(b) y=log[((8x+3)^2 * (x^3+5x^2-9x+1))/(7x+3)^5]

(c) Given f(x)=(4x)^10x^3, find f'(x)

(d) Differentiate f(x)=4x^10x^3, find f'(x)

please graph the function f(x)=(x-2)/(x-1) by finding

the domain
the x and y intercepts
the vertical asymptotes
the horizontal asymptotes
the intervals of increase and decrease
the local mins/max
the intervals of concavity
the inflection point(s) as an ordered pair

Solve the differential equation dy/dx = 15x + 5y/ 5x + 15y.

Write your solution without logarithms, and use a single, consolidated c as a constant

If a circle C with radius 1 rolls along the outside of the circle x² + y² = 36, a fixed point P on C traces out a curve called an epicycloid, with parametric equations x = 7 cos(t) − cos(7t), y = 7 sin(t) − sin(7t). Graph the epicycloid.

Find the area it encloses.

1. Find an equation for the line in the xy−plane that is tangent to the curve at the point corresponding to the given value of t. Also, find the value of d^2y/dx^2 at this point. x=sec t, y=tan t, t=π/6

2. Find the length of the parametric curve: x=cos t, y=t+sin t, 0 ≤ t ≤ π. Hint:To integrate , use the identity,  and complete the integral.

Suppose that a large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min. If the concentration of the solution entering is 2 lb/gal, determine a differential equation (in lb/min) for the amount of salt A(t) (in lb) in the tank at time

t > 0.

(Use A for A(t).)

solve the given boundary value problem.
y"+6y=24x, y(0)=0, y(1)+y'(1)=0