Calculus Unit 1

Try it now. Enter the interval equivalent to 2<x≤5 or x>7

(1.1) Answers: (2,5]U(7,oo)

Given the function f(x)=7x^2−3x+8 Calculate the following values: 1. f(-2) 2. f(-1) 3. f(0) 4. (1) 5. f(2)

(1.1) Answers: 1. 42 2. 18 3. 8 4. 12 5. 30

Given the function f(x)={3x+5 x<0 {3x+10 x≥0 Calculate the following values: f(-1) f(0) f(2)

(1.1) Answers: 2 10 16

Find the domain of f(x)=√4x−7

(1.1) Answers: [7/4,∞)

What is the domain of the following function: f(x)=√x+5x/x−3

(1.1) Answers: [−5,3)∪(3,∞)

Find the domain of the function f(x)=4x/x−3

(1.1) Answers: {x∣x≠3}

Find the equation (in terms of x) of the line through the points (-4,4) and (4,-3)

(1.2) Answers: y= −7/8x+1/2

Find the point at which the line f(x)=−x+3 intersects the line g(x)=5x+3

(1.2) Answers: (0,3)

In 1993, the moose population in a park was measured to be 4850. By 1999, the population was measured again to be 5930. If the population continues to change linearly: 1. Find a formula for the moose population, P, in terms of t, the years since 1990. 2. What does your model predict the moose population to be in 2003?

(1.2) Answers: 1. 180t+4310 2. 6650

A city's population in the year x=1963 was y=2,947,700. In 1969 the population was 2,946,500.Compute the slope of the population growth (or decline) and choose the most accurate statement from the following: The population is increasing at a rate of 200 people per year. The population is decreasing at a rate of 400 people per year. The population is decreasing at a rate of 300 people per year. The population is decreasing at a rate of 200 people per year. The population is increasing at a rate of 400 people per year. The population is increasing at a rate of 300 people per year.

(1.2) Answers: The population is decreasing at a rate of 200 people per year.

Find the slope of the line that goes through the points (9,-12) and (12,-14).

(1.2) Answers: m=−14−(−12)/12−(9) =−2/3

Find an equation of the line that (a) has the same y-intercept as the line y+5x+12=0 and (b) is parallel to the line −7x−10y=−3 Write your answer in the form y=mx+b

(1.2) Answers: y=-7/10x-12

Write an equation for a line perpendicular to y=−5x+3 and passing through the point (-10,0)

(1.2) Answers: y=1/5x+2

Find the equation of the line with Slope = -8 and passing through (-5,41). Write your equation in the form y=mx+b

(1.2) Answers: y=−8x+1

A culture of bacteria grows according to the continuous growth model B=f(t)=300e^(0.069t) where B is the number of bacteria and t is in hours. Findf(0) To the nearest whole number, find the number of bacteria after 5 hours. To the nearest tenth of an hour, determine how long it will take for the population to grow to 500 bacteria.

(1.3-1.4) Answers: 300 424 7.4

Certain radioactive material decays in such a way that the mass remaining after t years is given by the function m(t)=305e^(−0.02t) where m(t) is measured in grams (a) Find the mass at time t=0 (b) How much of the mass remains after 30 years?

(1.3-1.4) Answers: 305 167.4

A bacteria culture starts with 300 bacteria and grows at a rate proportional to its size. After 4 hours there will be 1200bacteria. (a) Express the population P after t hours as a function of t. Be sure to keep at least 4 significant figures on the growth rate. (b) What will be the population after 5 hours? (c) How long will it take for the population to reach 1810? Give your answer accurate to at least 2 decimal places.

(1.3-1.4) Answers: p(t)=300e^(0.34657t) 1697 5.1859105829494

Find the domain of y=log(4−5x)

(1.3-1.4) Answers: (−∞,4/5)

If e^5x=21, then x=

(1.3-1.4) Answers: 0.60890448754468

Find the solution of the exponential equation 15e^(x+2)=14 The exact solution (using natural logarithms) is: The approximate solution, rounded to 4 decimal places is:

(1.3-1.4) Answers: ln(14/15)−2 -2.069

Use the like-bases property and exponents to solve the equation (1/2)^(n+6)=2^(6n+8)

(1.3-1.4) Answers: n=-2

Use the like-bases property and exponents to solve the equation 8^n=256

(1.3-1.4) Answers: n=8/3

Solve for x. Use a calculator to round answer to 4 places after the decimal point. If ln(5x+3)=4

(1.3-1.4) Answers: x=10.319630006629

Solve for x: log5x+log5(x+4)=5

(1.3-1.4) Answers: x=53.937465083788

Let f(x)= 3.4x+28.4 if x<−6 √x+70 if x>−6 −2 if x=−6 Determine which one of the following rules for continuity is violated at x=−6 1. lim x→a f(x) exists. 2. f(a) is defined. 3. lim x→a f(x)=f(a) 4. None of the above; the function is continuous at x=−6x=-6.

(2.1) Answers: 3

Evaluate the limit: lim (z→ −3) 4z^4−2z

(2.1) Answers: 330

Evaluate the limit: lim(x→9) (−6x+54)/(x^2−11x+18)=

(2.1) Answers: −6/7

A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 31 ft/s. Its height in feet after t seconds is given by y=31t−20t^2 A. Find the average velocity for the time period beginning when t=2 and lasting .01 s: .005 s: .002 s: .001 s

(2.2) Answers: -49.2 -49.1 -49.04 -49.02

Let f(x)=5x^2+3x+2 Expand and simplify: f(x+h)−f(x)/h

(2.2) Answers: 10x+5h+3

Use the limit definition of the derivative to find the slope of the tangent line to the curve f(x)=3x^2+6x+2 at x=3

(2.2) Answers: 24

Suppose an arrow is shot upward on the moon with a velocity of 53 m/s, then its height in meters after t seconds is given by h(t)=53t−0.83t^2. Find the average velocity over the given time intervals. [2, 3]: [2, 2.5]: [2, 2.1]: [2, 2.01]: [2, 2.001]:

(2.2) Answers: 48.85 49.265 49.597 49.6717 49.67917

Let f(x) be the function 12x^2−7x+3. Then the quotient f(1+h)−f(1)/h can be simplified to ah+b a= b=

(2.2) Answers: a=12 b=17

Evaluate the limit: lim x→0 (8x)/((√6x+36)−6)

16