DIAGNOSTIC TESTS Success in calculus depends to a large etent on knowledge of the mathematics that precedes calculus: algebra, analtic geometr, functions, and trigonometr. The following tests are intended to diagnose weaknesses that ou might have in these areas. After taking each test ou can check our answers against the given answers and, if necessar, refresh our skills b referring to the review materials that are provided. A DIAGNOSTIC TEST: ALGEBRA. Evaluate each epression without using a calculator. 4 4 4. Simplif each epression. Write our answer without negative eponents. 5 5 s s a b 4ab. Epand and simplf. 6 4 5 (sa sb )(sa sb ) 6 4 4 5 4. Factor each epression. 4 5 4 9 6 5 4 7 4 5. Simplif the rational epression. 4 9 iv
DIAGNOSTIC TESTS v 6. Rationalize the epression and simplif. s s5 7. Rewrite b completing the square. 8. Solve the equation. (Find onl the real solutions.) 4 (g) 4 s4 9. Solve each inequalit. Write our answer using interval notation. 4 5 7 8. State whether each equation is true or false. p q p q sa b a b 5 4 4 s4 h h 4 4 sab sa sb TC T C a b a b ANSWERS TO DIAGNOSTIC TEST A: ALGEBRA. 8 8 5 9. 6s 48a 5 b 7. 4 7 5 a b 4 9 6 8 4. 5 5 4 9 5. 4 8 8 9 7 6. 5s s ( ) 4 7. 8. 6, 4 s, s, (g) 5 s4 h 7 9. 4,, 4,,, 7, 4. False True False False False True If ou have had difficult with these problems, ou ma wish to consult the Review of Algebra on the website www.stewartcalculus.com.
vi DIAGNOSTIC TESTS B DIAGNOSTIC TEST: ANALYTIC GEOMETRY. Find an equation for the line that passes through the point, 5 and has slope is parallel to the -ais is parallel to the -ais is parallel to the line 4. Find an equation for the circle that has center, 4 and passes through the point,.. Find the center and radius of the circle with equation 6 9. 4. Let A7, 4 and B5, be points in the plane. Find the slope of the line that contains A and B. Find an equation of the line that passes through A and B. What are the intercepts? Find the midpoint of the segment AB. Find the length of the segment AB. Find an equation of the perpendicular bisector of AB. Find an equation of the circle for which AB is a diameter. 5. Sketch the region in the -plane defined b the equation or inequalities. 4 and 4 9 6 44 ANSWERS TO DIAGNOSTIC TEST B: ANALYTIC GEOMETRY.. 4 5. Center, 5, radius 5 5 6 5. 4 4 _ =- 4. 4 4 6 ; -intercept 4, -intercept, 4 4 4 6 _ = - + =4 4 If ou have had difficult with these problems, ou ma wish to consult the Review of Analtic Geometr on the website www.stewartcalculus.com.
DIAGNOSTIC TESTS vii C DIAGNOSTIC TEST: FUNCTIONS FIGURE FOR PROBLEM. The graph of a function f is given at the left. State the value of f. Estimate the value of f. For what values of is f? Estimate the values of such that f. State the domain and range of f. f h f. If f, evaluate the difference quotient and simplif our answer. h. Find the domain of the function. f t s h s4 s 4. How are graphs of the functions obtained from the graph of f? f f f 5. Without using a calculator, make a rough sketch of the graph. 4 s s (g) (h) 6. Let f if if Evaluate f and f. Sketch the graph of f. 7. If f and t, find each of the following functions. f t t f t t t ANSWERS TO DIAGNOSTIC TEST C: FUNCTIONS..8,.5,.,,,. 6h h 4.,,,,,, 4 4. Reflect about the -ais Stretch verticall b a factor of, then shift unit downward Shift units to the right and units upward (g) _ (h) 5. _ (, ) 6., 7. f t 4 8 t f 4 5 t t t 8 _ If ou have had difficult with these problems, ou should look at Sections.. of this book.
viii DIAGNOSTIC TESTS D DIAGNOSTIC TEST: TRIGONOMETRY. Convert from degrees to radians. 8. Convert from radians to degrees. 56. Find the length of an arc of a circle with radius cm if the arc subtends a central angle of. 4. Find the eact values. tan sin76 sec5 4 a b FIGURE FOR PROBLEM 5 5. Epress the lengths a and b in the figure in terms of. 6. If sin and sec 5 4, where and lie between and, evaluate sin. 7. Prove the identities. tan sin cos sec tan sin tan 8. Find all values of such that sin sin and. 9. Sketch the graph of the function sin without using a calculator. ANSWERS TO DIAGNOSTIC TEST D: TRIGONOMETRY. 5. 5 6 4.6 6. 5 (4 6s ) 8.,,, 5,. cm 9. 4. s 5. 4 sin 4 cos _π π If ou have had difficult with these problems, ou should look at Appendi D of this book.