Calculus 5-R Unit 5 Logarithmic, Exponential and Other Transcendental Functions Review Problems

Find the domain of the function: f(x) = ln(3x + 1). Find the domain of the function: f(x) = 3 + ln(x - 1). (1,  ) 1

Review Problems Match the graph with the correct function [A] f(x) = ln x [B] f(x) = e x-1 [C] f(x) = ln(x - 1) [D] f(x) = e x 2

Review Problems Sketch the graph: f(x) = ln|x|. Solve for x: ln(5x + 1) + ln x = ln 4 3

Review Problems Solve for x: ln(5x - 1) - ln x = 3. dy/dx for y = ln(5 - x) 6 4

Review Problems Find the derivative: f(x) = ln(x 3 + 3x) 3 Find the derivative: 5

Review Problems Find the derivative: 6

Review Problems Differentiate: y = ln(ln tan x) Find y’ y = ln|2x2 - 5| 7

Review Problems Find y’ if ln xy = x + y Use logarithmic differentiation to find 8

Review Problems Find the slope of the tangent line to the graph of y = ln x 2 at the point where x = e 2 Evaluate the integral: ln 4 9

Review Problems Evaluate the integral: ln|ax + b| + C Evaluate the integral: -2 10

Review Problems Evaluate the integral: x + ln(x 2 + 1) + C 11

Review Problems Evaluate the integral: 8x + ln(x 2 + 1) + C Evaluate the integral: 9x - ln(x 2 + 1) + C 12

Review Problems Evaluate the integral: + C Evaluate the integral: ln|sec 3x| + C 13

Review Problems Evaluate the integral: -2 cos x + ln|csc x + cot x| + C Evaluate the integral: ln|tan x| + C 14

Review Problems Evaluate the integral: Differentiate: 15

Review Problems Match the graph shown with the correct function [A] f(x) = e (x-1) [B] f(x) = e -(x-1) [C] f(x) = e x + 1 [D] f(x) = e -x

Review Problems Differentiate: 17

Review Problems Find: if xe y + 1 = xy Evaluate the integral: + C 18

Review Problems Find the slope of the tangent line to the graph of y = (ln x)e x at the point where x = 2 Evaluate the integral: -e cosx + C 19

Review Problems Evaluate the integral: -95 e -t/5 + C Evaluate the integral: + C 20

Review Problems Find if y = 3 x x 3 3 x x 2 [3 + (ln 3)x] Differentiate: y = x 1-x x 1-x 21

Review Problems Differentiate y = x x x x [1 + ln x] Evaluate the integral: + C 22

Review Problems Find the area bounded by the function f(x) = 2 -x, the x-axis, x = -2, and x = 1 A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present 2 hours later, how many will there be 5 hours from the initial time given?

Review Problems A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present 2 hours later, how many hours (from the initial given time) will it take for the numbers to be 2500? Round your answer to 2 decimal places A mold culture doubles its mass every three days. Find the growth model for a plate seeded with 1.6 grams of mold. [Hint: Use the model y = Ce kt where t is time in days and y is grams of mold.] 1.6e t 24

Review Problems The balance in an account triples in 21 years. Assuming that interest is compounded continuously, what is the annual percentage rate? 5.23% The balance in an account triples in 20 years. Assuming that interest is compounded continuously, what is the annual percentage rate? 5.49% 25

Review Problems A radioactive element has a half-life of 50 days. What percentage of the original sample is left after 85 days? 30.78% A radioactive element has a half-life of 40 days. What percentage of the original sample is left after 48 days? 43.53% 26

Review Problems The number of fruit flies increases according to the law of exponential growth. If initially there are 10 fruit flies and after 6 hours there are 24, find the number of fruit flies after t hours. y = 10e ln(12/5)t/6 Determine whether the function y = 2cos x is a solution to the differential equation No 27

Review Problems verify that is a solution to the differential equation Find the particular solution to the differential equation given the general solution and the initial condition y = 1 - cos x 28

Find the particular solution to the differential equation given the general solution and the initial condition y(0) = 5. Review Problems Use integration to find a general solution to the differential equation y = (x + 1) 3/2 + C 29

Review Problems Use integration to find a general solution to the differential equation y = 3 ln|1 + x| + C Use integration to find a general solution to the differential equation y= 30

Review Problems Find the general solution to the first-order differential equation: (4 - x)dy + 2y dx = 0 y = C(4 - x) 2 Find the general solution to the first-order differential equation: x cos 2 y + tan y = 0 x 2 + sec 2 y = C 31

Review Problems Find the general solution to the first-order differential equation: y dx + (y - x)dy = 0 y ln|y| + x = Cy Find the general solution to the first-order differential equation: y= 32

Review Problems Find the general solution to the first-order differential equation: Find the particular solution of the differential equation that satisfies the initial condition y(0) = 7 y = e -x 33

Review Problems Find the solution to the initial value problem y(-1) = 0 e y + sin y = (x 2 + 1) Find the solution to the initial value problem y(1) = 0 ln(1 + y 2 ) = 2 ln x + x

Review Problems A deposit of $1000 is made into a fund with an annual interest rate of 5 percent. Find the time (in years) necessary for the investment to double if the interest is compounded continuously. Round your answer to 2 decimal places years A deposit of $1000 is made into a fund with an annual interest rate of 5 percent. Find the time (in years) necessary for the investment to triple if the interest is compounded continuously. Round your answer to 2 decimal places years 35

Review Problems Find the constant k so that the exponential function y = 3e kt passes through the points given on the graph. Solve the differential equation: 36

Review Problems Find the particular solution to the differential equation given the general solution and the initial condition y = 1 - cos x Find the general solution of the differential equation sin y + cos x = C 37

Review Problems Find the general solution of the differential equation Evaluate: arccos 38

Review Problems Find the exact value: 39

Review Problems Find the exact value: Find the exact value: sin(arctan 3) 40

Review Problems Find the exact value: 41

Review Problems Write an algebraic expression for tan[arcsin x]. Differentiate: 42

Review Problems Differentiate: y = arctan e x Differentiate: 43

Review Problems Differentiate: Differentiate: h(t) = arccos t 2 44

Review Problems Evaluate the integral: + C arcsec|2x| + C 45

Review Problems Evaluate the integral: ln(x 2 + 9) + arctan + C + C 46

Review Problems Evaluate the integral: + C arctan + C 47

Review Problems Evaluate the integral: arctan + C + C 48

Review Problems Evaluate the integral: arcsin(x - 2) + C + C 49

Review Problems Evaluate the integral: + C arcsec + C 50

Review Problems Solve the differential equation: y = + C y = Ce arcsec x 51

Answers 1. 2.C 3.Graph (1,  ) ln 4 ln|ax + b| + C -2 x + ln(x 2 + 1) + C 8x + ln(x 2 + 1) + C 9x - ln(x 2 + 1) + C + C ln|sec 3x| + C -2 cos x + ln|csc x + cot x| + C ln|tan x| + C

Answers D C -e cosx + C -95 e -t/5 + C + C 3 x x 2 [3 + (ln 3)x] x 1-x x x [1 + ln x] + C e t 5.23%5.49% 30.78%43.53% y = 10e ln(12/5)t/6 No y = 1 - cos x

Answers y = (x + 1) 3/2 + C y = 3 ln|1 + x| + C y= y = C(4 - x) 2 x 2 + sec 2 y = C y ln|y| + x = Cy y= y = e -x e y + sin y = (x 2 + 1) ln(1 + y 2 ) = 2 ln x + x years years y = 1 - cos x sin y + cos x = C

Answers C arcsec|2x| + C ln(x 2 + 9) + arctan + C + C arctan + C + C arcsin(x - 2) + C + C arcsec + C y = + C y = Ce arcsec x