Calculus Answers

find the slope of the tangent line to f(x)=x² + 2x at x=3

To show that d/dz(sin z) = cos z

Use the definition of limit to prove that the sequence {n −(1/n)}n=1 to infinity is divergent.

Let {an}∞n=1 be a bounded sequence and {bn}∞n=1 be a sequence converges to 0. Prove that the sequence {an · bn}∞n=1 converges to 0.

The position of an object moving along a line is given by the function s(t)=−15t2+75t. Find the average velocity of the object over the following intervals.

​(d) ​[1, 1+​h] where h>0 is any real number.

Solve the differential equation dy/dx = 2y+3e^x with x0 = 0, y0 = 0, using Taylor’s series method of order 2 to obtain the value of y at x = 0.1, 0.2.

Find the equation of the line passes through the point (3, −2) and is perpendicular to the line 3x − 2y = 4.

An environmental study of a certain community suggests that the average daily level of pollution in the air will be Q(p) = √ 0.6p + 20 units when the population is p thousand. It is estimated that after t years the population will be p(t) = 9 + 0.5t²  thousand.

(a) Express the level of pollution in the air as a function of time.

(b) compute the level of pollution after 5 years from now.

(c) When will the pollution level reach 10 units? 

1. Let f(x) = √ x + 1 − 1 4−x²  . Compute the following: (10 points)

i. f(0)

ii. f(-3)

iii. f(2)

iv. f(-1)

v. f(3)

vi. Domain(f(x))

2. (a) Let f( x x−2 ) = 3x + 4 find f(x).

(b) Compute difference quotient of the function g(x) = √ x² − 9, and simplify your answer.

3. Let f(x) = 1 − x, g(x) = x² + bx + c. find b and c such that fog(x) = −x²  + 5x + 4. + bx + c. find b and c such that fog(x) = −x²+ 5x + 4.

Use the definition of limit to prove that both of the sequences {1/ √ n } and { (−1)^n/ √ n } converges to 0.

Use the definition of limit to prove that both of the sequences { 1/√n } and { (−1)^n/√ n } converges to 0.