Tangent Lines and Linear Approximations Solutions
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1 Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment, based on the group of students, to determine the order and selection of questions to work in the session. Be sure to include a variety of types of questions (multiple choice, free response, calculator, and non-calculator) in the time allotted. Multiple Choice Answers:. C (003 AB6/BC6) Using the point-slope form of the equation of the tangent line and the point of tangency: y7 ( f())( x ) 7 ( f ())( ) 9 3 f(); f() 3.. C (997 AB0) y sin( x) When x, y cos 0 ; therefore, the equation of the tangent line is y0x. 3. E (003 AB89) g() f() 3 6 g( x) xf( x) f( x) (product rule); g() f() f() ( 5) 3 7. y67( x ) The only answer choice with slope of 7 is answer E.. A (008 AB) Using the product rule and factoring: 3 f ( x) 6 x( x) ( x) = ( x) ( 6x x) When x, f () 7 giving y 7( x ) The slope-intercept form of the equation of the tangent line is answer A. 5. B (993 AB7) (3x )() (x3)(3) Using the quotient rule, y. (3x ) When x, y() 3 so the point-slope form of the tangent line equation is y53( x ) which is equivalent to the standard form in answer B. Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
2 6. C (997 AB) The tangent line is y5( x 3). Approximate the zero when x 3. 5( x 3) 3 x B (997 AB) The slope of the line, y 3 x, is m. Given y x, the derivative is y x. Set x (slope of the parallel line y x 3 ). Substituting for x in the original equation, y. 8 Therefore, answer B gives the correct point, B (969 AB36/BC36) When x 0, y(0) sin(0) ( sin ) y x (cos x ) y(0) The equation of the tangent line is y ( x 0). Use this tangent line to approximate: y (0.) B (003 AB6) Use implicit differentiation: 6yy x(xy y). Evaluate at (3, ) : 6() y(3) ((3) y ()) 8y 8 y 9 Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
3 0. A (993 BC7) Using implicit differentiation: ln( xy) x ln x ln y x y x y Evaluate the original equation when x, ln( y) yields y e. Therefore, y e which yields the slope of the tangent line, y 0.. A (997 AB80) x f( x) 8 x( e ) The slope of the tangent line is equal to 3 when f( x) 3. Use the graphing calculator to x determine the x value when 8 x( e ) 3. Student may also use the derivative functions to graph f without calculating the derivative by hand.. D (998 AB87) Using the graphing calculator, determine the x value when f( x) and store the value in A Evaluate f( A) and store in B. The equation of the tangent line will be y B ( x A) or y x 0. *storing the values in this question is not necessary to obtain the correct answer, but is a good practice for students. 3. C (998 AB77) The functions f and g have parallel tangent lines when f ( x) g( x). x Students can take the derivatives by hand, 6e 8x, or use the calculator to solve. x Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
4 Free Response. 00 AB6ab dy (a) f 8 dx, : f : answer An equation of the tangent line is y 8 x. (b) f..8 Since y f x 0 on the interval x., d y 3 y 3x y 0 on this interval. dx Therefore on the interval x., the line tangent to the graph of y f x at x lies below the curve and the approximation f...8 is less than : approximation : conclusion with explanation Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
5 5. 00 AB6b y 5 x (b) f The approximation is less than f. because the graph of f is concave up on the interval x.. 3 : tangent line : computes y on the tangent line at x. : answer with reason Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
6 AB6 a (a) let Q bea, a 500 dy x dx 50 setting slopes equal: a a 0 a a a 00 or dy x dx 50 a equation for l: y x0 50 setting y-values equal: a a a0a a 00 3 (b) y x 0 5 (c) height of hill at x 50 : 50 y 50 5 feet 500 height of line at x 50 : y feet 5 Yes, the spotlight hits the tree since the height of the line is less than the height of the hill + tree which is 75 feet. : slope of tangent line from parabola : uses the condition that a a, a is on linel 500 : uses the condition that slopes are equal atq : answer 0 if student is solving an irrelevant equation : slope 0 if m 0 : equation : height of hill : height of line 3 0 if height 0 : answer with analysis Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
7 7. 005B AB5/BC 5 (a) yy y xy yx y y y y y x : implicit differentiation : solves for y (b) y y x y y x x 0 y 0,, 0, y : y x : answer y (c) 0 y x y 0 The curve has no horizontal tangent since 0 x 0 for any x. : 0 y : explanation Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
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