Mathematics for the International Student: Mathematics SL 3rd Edition solutions

Mathematics for the International Student: Mathematics SL 3
Find the equation of the tangent to: a y = x 2x2 + 3 at x = 2 b y = px + 1 at x = 4 c y = x3 5x at x = 1 d y = 4 px at (1, 4) e y = 3 x 1 x2 at (1, 4) f y = 3x2 1 x at x = 1.

Mathematics for the International Student: Mathematics SL 3
Find the equation of the normal to: a y = x2 at the point (3, 9) b y = x3 5x + 2 at x = 2 c y = 5 px px at the point (1, 4) d y = 8px 1 x2 at x = 1.

Mathematics for the International Student: Mathematics SL 3
a Find the equations of the horizontal tangents to y = 2x3 + 3x2 12x + 1. b Find the points of contact where horizontal tangents meet the curve y = 2px + 1 px . c Find k if the tangent to y = 2x3 + kx2 3 at the point where x = 2 has gradient 4. d Find the equation of the other tangent to y = 1 3x

Mathematics for the International Student: Mathematics SL 3
Find the equation of the tangent to: a y = p2x + 1 at x = 4 b y = 1 2 x at x = 1 c f(x) = x 1 3x at (1, 1 4 ) d f(x) = x2 1 x at (2, 4).

Mathematics for the International Student: Mathematics SL 3
Find the equation of the tangent to: a y = p2x + 1 at x = 4 b y = 1 2 x at x = 1 c f(x) = x 1 3x at (1, 1 4 ) d f(x) = x2 1 x at (2, 4).

Mathematics for the International Student: Mathematics SL 3
The curve y = a p1 bx where a and b are constants, has a tangent with equation 3x + y = 5 at the point where x = 1. Find a and b.

Mathematics for the International Student: Mathematics SL 3
The curve y = a p1 bx where a and b are constants, has a tangent with equation 3x + y = 5 at the point where x = 1. Find a and b.

Mathematics for the International Student: Mathematics SL 3
Find the equation of: a the tangent to the function f : x 7! ex at the point where x = 1 b the tangent to y = ln(2 x) at the point where x = 1 c the normal to y = ln px at the point where y = 1.

Mathematics for the International Student: Mathematics SL 3
Show that the curve with equation y = cos x 1 + sin x does not have any horizontal tangents.

Mathematics for the International Student: Mathematics SL 3
Find the equation of: a the tangent to y = sin x at the origin b the tangent to y = tan x at the origin c the normal to y = cos x at the point where x = 6 d the normal to y = 1 sin(2x) at the point x = 4 .

Mathematics for the International Student: Mathematics SL 3
a Find where the tangent to the curve y = x3 at the point where x = 2, meets the curve again. b Find where the tangent to the curve y = x3 + 2x2 + 1 at the point where x = 1, meets the curve again.

Mathematics for the International Student: Mathematics SL 3
Consider the function f(x) = x2 + 4 x2 . a Find f0 (x). b Find the values of x at which the tangent to the curve is horizontal. c Show that the tangents at these points are the same line.

Mathematics for the International Student: Mathematics SL 3
The tangent to y = x2ex at x = 1 cuts the x and y-axes at A and B respectively. Find the coordinates of A and B.

Mathematics for the International Student: Mathematics SL 3
a Find the equation of the tangent to y = x2 x + 9 at the point where x = a. Hence, find the equations of the two tangents from (0, 0) to the curve. State the coordinates of the points of contact. b Find the equations of the tangents to y = x3 from the external point (2, 0). c Find the equation of t

Mathematics for the International Student: Mathematics SL 3
Find the equation of the tangent to y = ex at the point where x = a. Hence, find the equation of the tangent to y = ex which passes through the origin.

Mathematics for the International Student: Mathematics SL 3
Consider f(x) = 8 x2 . a Sketch the graph of the function. b Find the equation of the tangent at the point where x = a. c If the tangent in b cuts the x-axis at A and the y-axis at B, find the coordinates of A and B. d Find the area of triangle OAB and discuss the area of the triangle as a ! 1.

Mathematics for the International Student: Mathematics SL 3
Find, correct to 2 decimal places, the angle between the tangents to y = 3ex and y =2+ ex at their point of intersection.

Mathematics for the International Student: Mathematics SL 3
A quadratic of the form y = ax2, a > 0, touches the logarithmic function y = ln x as shown. If two curves touch then they share a common tangent at that point. a If the x-coordinate of the point of contact is b, explain why ab2 = ln b and 2ab = 1 b . b Deduce that the point of contact is ( pe, 1 2 ).