代写MATH 210: STUDY GUIDE FOR MIDTERM 1代做回归

MATH 210: STUDY GUIDE FOR MIDTERM 1

Here are the big things you should know for the exam.

• How to do implicit differentiation.

• How to do logarithmic differentiation.

• How to use the inverse function rule to compute derivatives.

• How to use the rules for derivatives to compute derivatives.

• How to derive the rules for derivatives, whether from the definition of the derivative or from other rules.

• How to find the equation for a tangent line.

• How to calculate the standard part of an expression involving infinitesimal and infinite numbers.

• How to determine whether an expression gives an infinitesimal, finite but not infini-tesimal, or infinite number.

For the exam you will get a formula sheet with some but not all of the rules we have learned. I will post the formula sheet once it is finalized.

Here’s some sample problems to practice for the exam.

(1) Write an equation for the line tangent to the curve exy − e2y = 0 at the point (2, 1).

(2) What are the slopes of the ellipse 16x 2 + 9y 2 = 25 at the points (±1, ±1)?

(3) Differentiate a(x) = xsinx.

(4) Use logarithmic differentiation to differentiate b(x) = x 3 e 2x cos x.

(5) Use logarithmic differentiation, the rule for the derivative of ln x, and the chain rule to derive the quotient rule.

(6) Use the other rules for the derivative to derive the rules for tan x, cot x, sec x, and csc x.

(7) Use the other rules for the derivative to derive the rules for b x , ln x, and logb x.

(8) Use the definition of the derivative in terms of standard systems to compute the derivative of x 2 − x.

(9) Use the definition of the derivative in terms of standard systems to compute the derivative of 1/ √x.

(10) Differentiate a(x) = e x cos x. What is a 0 (0)?

(11) Find the first and second derivatives of b(t) = 5000 + t − 10t 5 .

(12) Differentiate c(x) = √ 1 + ln x. What is c 0 (1)? Give an equation for the line tangent to y = c(x) at x = 1.

(13) Differentiate d(x) = arcsin(tan(2x)).

(14) Differentiate f(x) = x2+e/ex2+e.

(15) Differentiate g(x) = arctan(x + csc x).

For the following problems, ε and ∆x are nonzero infinitesimals, H is positive and infinite, and x is a real number.

(16) What is st(5x + x∆x − 3∆x 2 )?

(17) What is

st (2H4 + 100/H2 − H4)?

(18) What is st(√H + 1 − √H − 1)?

(19) Determine whether the following is infinitesimal, finite but non-infinitesimal, or infi-nite:

3ε3/ε2 + 2ε + 1.

(20) Determine whether the following is infinitesimal, finite but non-infinitesimal, or infi-nite:

3H3 + 1/H2 − 1000.

(21) Determine whether the following is infinitesimal, finite but non-infinitesimal, or infi-nite:

ε/3 + ε/3 + ε3ε2