MAT 265 for Engineers

LIMITS

100

lim_x->5 (2x² - 3x + 4)

39

200

lim_x->-2 [(x³+2x²-1) / (5-3x)]

-1/11

300

lim_t->0  [((t²+9)^1/2 -3) / t²]

1/6

400

Evaluate lim_x->1 arcsin((1-(x)^1/2) / (1-x))

pi/6

500

lim_x->inf ((3x²-x-2) / (5x²+4x+1))

3/5

Grab Bag

100

Derivative of f(x) = cot(x)

f(x) = -csc²(x)

200

Integral of f(x) = tan(x)

ln|sec(x)| + C

300

Differentiate f(x) = [(sec(x)/(1+tan(x))]

f'(x) = secx(tanx-1) / (1+tanx)²

400

Find lim_x->0 [sin(x)/x]

1

500

Find the definite integral of [ln(x) / x]dx on the interval [1,e].

1/2

MISCELLANEOUS

100

Find the derivative of the function f(x) = x²-8x+9 at number "a" using the definition of a limit.

f'(a) = 2a - 8

200

Derivative of f(x) = 7e^7x

49e^7x

300

Find the n^th derivative of f(x) = xe^x

f⁽ⁿ⁾(x) = (x+n)e^x

400

Differentiate f(x) = (2x+1)⁵(x³-x+1)⁴

f'(x) = 2(2x+1)⁴(x³-x+1)³(17x³+6x²-9x+3)

500

Find the indefinite integral of f(x) = (1+x²)^1/2 x⁵ dx

1/7(1+x²)^7/2 - 2/5(1+x²)^5/2 + 1/3(1+x²)^3/2 + C

LOGARITHMS &EXPONENTS

100

The steps to logarithmic differentiation

1. Take the natural log of both sides of an equation

y = f(x) and use the Laws of Logarithms to simplify.

2. Differentiate implicitly with respect to x

3. Solve for the resulting equation y'.

200

Differentiate y = ln(x³ + 1)

y' = 3x² / (x³ + 1)

300

The definition of e

e = lim_x->0 (1 + x)^1/x

400

Differentiate y = x^{(x)1/2(power)}

y' = x^{(x)1/2(power)}((2+lnx)/(2x^1/2)

500

Differentiate 

y = (x^3/4(x²+1)^1/2) / (3x+2)⁵

y' = [(x^3/4(x²+1)^1/2) / (3x+2)⁵]*[(3/4x)+(x/(x²+1)) - (15/(3x+2))

APPLICATIONS OFDIFFERENTIATION

100

Find the velocity at time t for the position function 

s = f(t) = t³ - 6t² + 9t

v(t) = 3t² - 12t + 9

200

When is the particle in the position function 

s = f(t) = t³ - 6t² + 9t

moving forward or the positive direction?

t < 1 and t > 3

(-inf,1) U (3,inf)

300

Find the total distance traveled by the particle in the function 

s = f(t) = t³ - 6t² + 9t

during the first five seconds.

28m

400

A bottle of soda pop at room temperature (72F) is placed in a refrigerator where the temperature is 44F. After half an hour, the soda pop has cooled to 61F. 

What is the temperature of the soda pop after another half hour and how long does it take for the soda pop to cool to 50F? 

dT/dt = k(T - 44) ~ Newton's Law of Cooling

~54.3 F 

~92.6 min

500

A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight.

0.128 rad/s