AP Calculus AB Summer Assignment for 2017-2018 School Year Mrs. Brennan In order to be prepared for next year and be ready to move on to new work, you must have skills required to do these problems with mastery. The answers are included so that you can check your work. Please do the work neatly on loose leaf and where appropriate, graph paper. Label work with problem number. Bring it in on the first day of class next year. It will be collected during the first week of school. Be sure that you show all your work. All graphs must include at least 3 points. Use tutorial.math.lamar.edu for extra help (Especially the Trig Sheet). Enjoy your summer. Simplify without a calculator: 1.) (a) 8 ' (b) ( 8) ' (c) 8 * ' Find the exact value without a calculator. 2.) (a) log 16 (b) log 0 ' (c) log 1 4 (d) log 3 3 Use logarithm properties for expanding to rewrite the expression in terms of r, s, and t, where r = ln a, s = ln b, and t = ln c. 3.) (a) ln a bc (b) ln? @ A B Expand the logarithm in terms of sums, differences, and multiples of simpler logarithms. 4.) (a) logc10x x 3F (b) ln GH IJK A G G H L0 Rewrite the expression as a single logarithm. 5.) 2 ln(x + 1) + 0 ln x ln(cos x) '
Solve for x without a calculator. 6.) log 0R C xf = 1 7.) ln T 0 G U = 2 8.) log (5 G ) = 8 9.) log 0R x ' log 0R x = 5 10.) ln T 0 G U + ln(2x' ) = ln 3 Solve for x without a calculator. Use the natural logarithm anywhere logarithms are needed. 11.) 5 *G = 3 12.) 2e 'G = 7 13.) xe *G + 2e *G = 0 Express in interval notation 14.) (a) {x: x 4} (b) {x: x > 4} Solve the inequality and sketch the solution on a coordinate line. 15.) 4 + 5x 3x 7 16.) 3 4 2x < 7 G 17.) < 4 G*' 'GL0 18.) G* 19.) x > 9 20.) (x 4)(x + 2) > 0 21.) x 9x + 20 0
Find all values of x for which the given expression yields a real number. 22.) x + x 6 Find the exact values of all six trigonometric functions of θ. 23.) The θ is an acute angle of a right triangle. Solve the problem by drawing an appropriate right triangle. Do not use a calculator. 24.) Find the length of the side adjacent to θ given that the hypotenuse has length 6 and cos θ = 0.3. Use the information to find the values of the remaining 5 trigonometric functions of θ. 25.) (a) cos θ = ', ` < θ < 0 (b)tan θ = 0, ` < θ < π ' Find all values of θ (in radians) that satisfy the given equation. Do not use a calculator. 26.) (a) tan θ = 1 (b) cos θ = 0 27.) (a) tan θ = 0 ' (b) sin θ = '
Find the values of all six trigonometric functions of θ. 28.) Find all values of θ (in radians) that satisfy the conditions. 29.) (a) sin θ = 0 (b) cos θ = 0 (c) tan θ = 0 (d) csc θ is undefined (e) sec θ is undefined (f) cot θ is undefined Do not use a calculator. 30.) If cos θ = ` and 0 < θ <, find ' (a) sin 2θ (b) cos 2θ For #31-34, sketch the graph of the equation. State the domain and range. 31.) y = 4 x 32.) y = x 4 33.) x x + y = 0 34.) xy = 1 35.) List the lines in the accompanying figure in order of increasing slope.
36.) A particle, initially at (1,2), moves along a line of slope m = 3 to a new position (x, y). (a) Find y if x = 5. (b) Find x if y = 2. Use the graph to find the equation of the line in slope-intercept form. 37.) (a) (b) For # 38-40, find the slope-intercept form of the line satisfying the given conditions. 38.) The line is perpendicular to y = 5x + 9 and its y-intercept is 6. 39.) The y-intercept is 2 and the x-intercept is 4. 40.) The line is perpendicular to the y-axis and passes through ( 4,1). 41.) For what value of k will the line 3x + ky = 4 (a) have slope 2 (b) have y-intercept 5 (c) pass through the point ( 2,4) (d) be parallel to the line 2x 5y = 1 (e) be perpendicular to the line 4x + 3y = 2 42.) Sketch the graph of y = 3x and explain how this graph is related to the graphs of y = 3x and y = 3x.
43.) Find (a) the distance between the points and (b) the midpoint of the line segment joining the points. A(7,1), B(1,9) 44.) Find the equation of the line that is a perpendicular bisector of the line segment joining (2,8) and ( 4,6). 45.) Find the center and radius of each circle. (a) x + y = 25 (b) (x 1) + (y 4) = 16 (c) (x + 1) + (y + 3) = 5 (d) x + (y + 2) = 1 For # 46-48, find the standard equation of the circle satisfying the given conditions. 46.) Center (3, 2); radius = 4 47.) Center ( 4,8); circle is tangent to the x-axis. 48.) Center ( 3, 4); circle passes through the origin. 49.) Find and equation of the bottom half of the circle x + y = 16. 50.) Graph y = 25 x. 51.) Find an equation of the line that is tangent to the circle x + y = 25 at the point (3,4) on the circle. 52.) Graph (a) y = x + 5 and (b) x = k4 y. Solve the inequality 53.) (a) 2x + 5x 1 < 0 (b) x 2x + 3 > 0
Answers 1.) (a) 4 (b) 4 (c) 0 1 2.) (a) 4 (b) 5 (c) 1 (d) 0 3.) (a) 2r + l + m (b) s 3r t 4.) (a) 1 + log x + 0 log(x 3) (b) 2 ln x + 3 ln sin x 0 ln(x + 1) A 5.) ln G (GL0) H opi G 6.) 0.01 7.) e 8.) 4 9.) 10 10.) q ' 11.) sk ' sk 0 12.) ln ' Tt U 13.) 2 14.) (a) [ 2,2] (b) (, 2) (2, ) 15.) (, 00 y and graph. 16.) T ', 0 y and graph. 17.) (, 3) (4, ) and graph. 18.) T ', 2U and graph.
19.) (, 3) (3, ) and graph. 20.) (, 2) (4, ) and graph. 21.) [4,5] and graph. 22.) (, 3] [2, ) 23.) sin θ = 0 ; cos θ = ; cot θ = 0 24.) 1.8 ; tan θ = 0 ; csc θ = 0 ; sec θ = 25.) (a) sin θ = 1 ; cos θ = ' ; tan θ = 1 ; csc θ = ; sec θ = ' 1 ' ; cot θ = ' 1 (b) sin θ = 0 ' ; cot θ = 3 26.) (a) ± nπ, n = 0,1,2, '`1 ' ; cos θ = ; tan θ = 0 ; csc θ = 2; sec θ = ' (b) ` ± 2nπ and ± 2nπ, n = 0,1,2, ' `' 27.) (a) ` ± nπ, n = 0,1,2, (b) ± 2nπ and ± 2nπ, n = 0,1,2, 1`' `' 28.) sin θ = ; cos θ = 0 ; tan θ = ; csc θ = ; sec θ = 0 0 ; cot θ = 0 29.) (a) θ = ±nπ, n = 0,1,2, (b) θ = ` ± nπ, n = 0,1,2, (c) θ = ±nπ, n = 0,1,2, (d) θ = ±nπ, n = 0,1,2, (e) θ = ` ± nπ, n = 0,1,2, (f) θ = ±nπ, n = 0,1,2, 30.) (a) 1 (b) 0 3 3 31.) Graph 32.) Graph
33.) Graph 34.) Graph 35.) III<II<I<I 36.) (a) 14 (b) 0 ' 37.) (a)y = ' x 3 (b) y = ' 1 x 38.) y = 0 x + 6 39.) y = 0 x + 2 40.) y = 1 41.) (a) ' (b) 1 (c) (d) 0 (e) 4 42.) Graph. It is the combination of the other 2 graphs. 43.) (a) 10 (b) (4,5) 44.) y = 3x + 4 45.) (a) (0,0); 5 (b) (1,4); 4 (c) ( 1, 3); 5 (d) (0, 2); 1 46.) (x 3) + (y + 2) = 16 47.) (x + 4) + (y 8) = 64 48.) (x + 3) + (y + 4) = 25 49.) y = 16 x 50.) Graph 51.) y = ' x + 1 1 52.) Graph 53.) (a) C** ''F 1 < x < C*L ''F 1 (b) < x <