1. Find the equation of the tangent line to the graph of ?(?) = 1 + ? + ???? at ? = 0. 4.

2. Find the equation of the tangent line to the graph of ?(?) = (?+1)/ (?−1)at ? = 0. 5.

Consider the spiral path x(t) = (cos^2t,sin^2t,t) for 0 ≤ t ≤ π/2. Evaluate the integral x dx−y dy + z^2 dz

P = 2xy + ? 2 + ??, ? = 2xy + ? 2 + ??, ? = ?? olacak şekilde bir vektör alanı veriliyor. A (1,2,3) ve B (2,1,5) olmak üzere şayet varsa verilen vektör alanının potansiyel fonksiyonunu bulunuz. Bu vektör alanında ∫ ??? ? ? + ??? + ??? eğrisel integralinin değerini hesaplayınız.

find a potential function and calculate the integral

Find the cubic equation:

f(x) = ax^3+bx^2+cx+d

for which f(-1)=3, f(1)=1, f(2)=6, and f(3)=7.

Find the value of a, b, c, and d

1) A mixture of 5 pounds of fertilizer A, 13 pounds of fertilizer B, and 4 pounds of fertilizer C provides the optimal nutrients for a plant. Commercial brand X contains equal parts of fertilizer B and fertilizer C. Commercial brand Y contains one part of fertilizer A and two parts of fertilizer B. Commercial brand Z contains two parts of fertilizer A, five parts of fertilizer B, and two parts of fertilizer C. How much of each fertilizer brand is needed to obtain the desired mixture?

2) A chemist needs 10 liters of a 25% acid solution. The solution is to be mixed from three solutions whose concentrations are 10%, 20% and 50%. How many liters of each solution will satisfy each condition?

a) Use 2 liters of the 50% solution.

b) Use as little as possible of the 50% solution.

c) Use as much as possible of the 50% solution.

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 120°.

(a) Find the rate of change of T at (1, 2, 2) in the direction toward the point (3, 3, 4).

True and False (No need to solve).

1. Every bounded continuous function is integrable.

2. f(x)=|x| is not integrable in [-1, 1] because the function f is not differentiable at x=0.

3. You can always use a bisection algorithm to find a root of a continuous function.

4. Bisection algorithm is based on the fact that If f is a continuous function and f(x₁) and f(x₂) have opposite signs, then the function f has a root in the interval (x₁, x₂). As a consequence, one can infer that if a continuous function f(x) has a root at x=a, then there exists a number h such that f(a- h) and f(a+ h) have opposite sign.

5. The total areas of step function, f(k)=k feet, for k ranging from 1 to 1000 is 500.50 ft², assuming each step has a width of 1 foot.

6. if f is continuous then d/dx (integral(from a to x) f(t)dt= f(x).

7. The total areas of step function, f(k)=k feet, for k ranging from 1 to 1000 is 550 ft², assuming each step has a width of 1 foot.

For x ∈ [−2, 6],h(x) = 2x, j(x) = x2, and k(x) = 2x.

1. Represent each function with a sample table

2. Graph all three functions on the same coordinate system

3. Find the average rate of change for each function at x = . -1, x = 2, and x = 4

Exercise 4: Find all local extrema for f ? = sin^2 ? + cos ? on the interval [− ?/ 2 , ?/ 2 ]. Verify your answer by graphing.

I need a partial differential EQUATION to govern the pressure change for a steady state fluid inside a horizontal pipe.

please, make some clarafication if possible.

thanks.

Write z = -6 + 6i in polar form.

Write z = 9 - 3sqrt3i in polar form.

Write z = 2 (cos 5pi/6 + i sin 5pi/6) in rectangular form.

Convert (6, 5pi/6) to exact rectangular coordinates.

Then give another ordered pair with a negative r and negative theta for the point (6, 5pi/6)

Use reference angles and symmetry on the unit circle to find the exact value of each expression. Do not use calculator.

(a) sin(135 degrees)

(b) cos(11pi/6)

(c) tan(-5pi/3)

(d) sec(-120 degrees)

(e) cot(5pi/2)