ARE YOU READY FOR CALCULUS?? Name: Date: Period:
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1 ARE YOU READY FOR CALCULUS?? Name: Date: Period: Directions: Complete the following problems. **You MUST show all work to receive credit.**(use separate sheets of paper.) Problems with an asterisk (*) are challenge problems. Try your best! 1. Simplify: 9 71 (b) (d) 9 1. Rationalize the denominator: (b) * Write each of the following epressions in the form real numbers. p q ca b where c, p, and q are a b (b) a 9ab b a ab a (d) b b (e) b a 1 1 a (f) a b 1 1 b a 4. Solve for. (Do not use a calculator): (b) log (d) log log 4 4log 5 5. Simplify: log 5 log 1log 1 (b) log49 log log5 6. Simplify: 1 1 log1 1 (b) log1 1 log log Solve the following equations for the indicated variables: y z 1, for a (b) V abbc ca, for a a b c * Ar rh, for positive r (d) A P nrp, for P (e) yd y d, for d (f) 1, for 4 Page 1 of 6
2 8. For the following equations, complete the square and reduce to one of the standard forms y b A a or a A y b. y 4 (b) y 9y 6y9 9. Factor completely: (b) (d) 1 1. Find all real solutions to: (b) Solve for : sin cos ; (b) cos sin sin ; tan sec cos ; 1. Without using a calculator, evaluate the following: 5 cos 1 1 (b) sin tan (d) sin 1 (e) 9 cos 4 (f) sin 1 (g) 7 tan 6 1 (h) cos 1 1. Given the graph of f sin, sketch the graphs of: f 4 sin (d) f cos (e) f (b) sin f 1 sin f sin f sin : 14. Solve the equations: 4 1 (b) Find the remainders on division of: by (b) 4 by The equation 1 has a solution. Find all other solutions. (b) Solve for, the equation (All solutions are rational and between 1. Page of 6
3 17. Solve the inequalities: *(b) Solve for : 5 8 (b) Determine the equations of the following lines: the line through 1, and, 4 ; (b) the line through 1, and perpendicular to the line y5 ; the line through, and the midpoint of the line segment from 1, 4 to,.. Find the point of intersection of the lines: y7 and 5y. (b) Shade the region in the y plane that is described by the inequalities: y7 5y 1. Find the equations of the following circles: the circle with center at 1, that passes through the point, 1 ; (b) the circle that passes through the origin and has intercepts equal to 1 and on the - and y-aes respectively.. For the circle y 64y, find the center and the radius; (b) the equation of the tangent at,5.. *A circle is tangent to the y-ais at y and has one -intercept at 1. Determine the other -intercept. (b) Deduce the equation of the circle. 4. A curve is traced by a point P, y which moves such that its distance from the point A 1,1 is three times the distance from the point, 1. equation of the curve. B Determine the Page of 6
4 5. Find the domain of the function: 1 f (b) Find the domain and range of the functions: 5 i) f 7 ii) g 1 6. Let f. Show that f f. 1,. 1, Find the domain and range of 7. Simplify:, f h f h where f (b) f 1 1 f 8. The graph of the function y f is given as follows. Determine the graphs of the functions: f 1 (b) f f (d) f 9. Sketch the graphs of the functions: g h (b) 1. The graph of a quadratic function (a parabola) has intercepts 1 and and a range consisting of all numbers less than or equal to 4. Determine an epression for the function. (b) Sketch the graph of the quadratic function y Write as a single equation in and y : t 1 t 1 (b) y t t y t t sin t y cost Page 4 of 6
5 . Find the inverse of the functions: 5 1 f (b) f * f 1,. A function f has the following graph. Sketch the graph of the inverse 1 function f. 4. Epress in terms of the other variables in the picture. 5. Find the ratio of the area inside the square but outside the circle to the area of the square in the picture below. (b) Find a formula for the perimeter of a window of the shape in the picture (b) above. A water tank has the shape of a cone (like an ice cream cone without ice cream.) The tank is 1m high and has a radius of m at the top. If the water is 5m deep (in the middle) what is the surface area of the top of the water? Page 5 of 6
6 (d) Two cars start moving from the same point. One travels south at 1 km/hour, the other west at 5 km/hour. How far apart are they two hours later? (e) A kite is 1m above the ground. If there are m of string out, what is the angle between the string and the horizontal? (Assume that the string is perfectly straight.) 6. You should know the following trigonometric identities: (A) sin sin (B) cos y cos cos y sin sin y cos cos sin y sin cos y cos sin y (C) (D) Use these to derive the following important identities, which you should also know: (e) sin cos 1 (b) sin sin cos cos cos sin (d) cos cos 1 cos 1 sin Page 6 of 6
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