Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 3x) / x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 7x) / 3x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 5x) / x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂2 (sin (x-2)) / (x² - 4)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.

Find the derivative of the following functions.

y = sin x + cos x

Find the derivative of the following functions.

y = e^-x sin x

Find the derivative of the following functions.

y = x sin x

Find the derivative of the following functions.

y = cos x/sin x + 1

23–51. Calculating derivatives Find the derivative of the following functions.

y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants

23–51. Calculating derivatives Find the derivative of the following functions.

y = cos² x

23–51. Calculating derivatives Find the derivative of the following functions.

y = sin x / 1 + cos x

23–51. Calculating derivatives Find the derivative of the following functions.

y = tan x + cot x

Find the derivative of the following functions.

y = cot x / (1 + csc x)

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.​

Find the velocity of the oscillator, v(t) =y′(t).

Find y'' for the following functions.

y = x sin x

Find y'' for the following functions.

y = e^x sin x

Find y'' for the following functions.

y = tan x

Find y'' for the following functions.

y = cos θ sin θ

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.

d/dx (sec x) = sec x tan x

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.

d/dx (csc x) = -csc x cot x

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/2 cos x/x−π/2

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/4 cot x−1 / x−π/4

Find an equation of the line tangent to the following curves at the given value of x.

y = csc x; x = π/4

How is lim x🠂0 sin x/x used in this section?

For what values of x does g(x) = x−sin x have a slope of 1?

Use a graphing utility to plot the curve and the tangent line.

y = cos x / 1−cos x; x = π/3

Explain why or why not Determine whether the following statements are true and give an explanation or counter example.

b. d²/dx² (sin x) = sin x.

A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.

b. Show that y = B cos t satisfies the equation for any constant B.