Calculate the product of each radical expression and its corresponding conjugate.
a. \displaystyle
\sqrt{5} - \sqrt{2}
b. \displaystyle
3\sqrt{5} + 2\sqrt{2}
c. \displaystyle
9 + 2\sqrt{5}
d. \displaystyle
3\sqrt{5} -2\sqrt{10}
Rationalize each denominator.
\displaystyle
\frac{5 +\sqrt{2}}{\sqrt{3}}
Rationalize each denominator.
\displaystyle
\frac{2\sqrt{3}+ 4}{\sqrt{3}}
Rationalize each denominator.
\displaystyle
\frac{5}{\sqrt{7} -4}
Rationalize each denominator.
\displaystyle
\frac{2\sqrt{3}}{\sqrt{3} -2}
Rationalize each denominator.
\displaystyle
\frac{5\sqrt{3}}{2\sqrt{3} + 4}
Rationalize each denominator.
\displaystyle
\frac{3\sqrt{2}}{2\sqrt{3} - 5}
Rationalize each numerator.
\displaystyle
\frac{\sqrt{2}}{5}
Rationalize each numerator.
\displaystyle
\frac{\sqrt{3}}{6 +\sqrt{2}}
Rationalize each numerator.
\displaystyle
\frac{\sqrt{7}-4}{5}
Rationalize each numerator.
\displaystyle
\frac{2\sqrt{3} -5}{3\sqrt{2}}
Rationalize each numerator.
\displaystyle
\frac{\sqrt{3} - \sqrt{7}}{4}
Rationalize each numerator.
\displaystyle
\frac{2\sqrt{3} + \sqrt{7}}{5}
Determine the equation of the line described by the given information.
slope - \frac{2}{3}, passing through the point (0, 6).
Determine the equation of the line described by the given information.
passing through points (2, 7) and (6, 11).
Determine the equation of the line described by the given information.
parallel to y = 4x - 6, passing through point (2, 6)
Determine the equation of the line described by the given information.
perpendicular to y = -5x + 3, passing through point (-1, -2).
Find the slope of PQ, in simplified form, given P(1, -1) and Q(1 + h, f(1 + h)), where f(x) = -x^2.
Consider the function y = x^2 -2x -2.
a) Copy and complete the following tables of values. P and Q are points on the graph of f(x).
b) Use your results to approximate the slope of the tangent to the graph of f(x) at point P.
c) Calculate the slope of the secant where the x-coordinates of Q is -1 +h.
d) Use your results for part c to calculate the slope of the tangent to the graph of f(x) at point P.
e) Compare your answers for parts b and d.
Calculate the slope of the tangent to each curve at the given point or value of x.
\displaystyle
f(x) = x^2 + 3x - 5, (-3, -5)
Calculate the slope of the tangent to each curve at the given point or value of x.
\displaystyle
f(x) = \frac{1}{x} , x= \frac{1}{3}
Calculate the slope of the tangent to each curve at the given point or value of x.
\displaystyle
f(x) = \frac{4}{x -2}, (6, 1)
Calculate the slope of the tangent to each curve at the given point or value of x.
\displaystyle
f(x) = \sqrt{x + 4}, x = 5
The function s(t) = 6t(t + 1) describes the distance (in km) that a car has travelled after a time t(in hours), for 0 \leq t \leq 6.
a) Calculate the average velocity of the car during the following intervals.
- i. from t =2 to t =3
- ii. from t =2 to t =2.1
- from t =2 to t = 2.01
b) Use your results from above to approximate the instantaneous velocity of the car when t =2.
c) Find the average of the car from t =2 to t = 2+ h.
d) Use your results for part c to find the velocity when t = 2.
Calculate the instantaneous rate of change of f(x) with respect to x at the given value of x.
\displaystyle
f(x) = 5- x^2, x = 2
Calculate the instantaneous rate of change of f(x) with respect to x at the given value of x.
\displaystyle
f(x) = \frac{3}{x}, x = \frac{1}{2}
An oil tank is being drained for cleaning. After t minutes, there are V litres of oil left in the tank, where V(t) = 50(30 -t)^2, 0 \leq t\leq 30.
a. Calculate the average rate of change in the volume during the first 20 mins
b. Calculate the rate of change in volume at time t = 20.
Find the equation of the tangent at the given value of x.
\displaystyle
y = x^2 + x - 3, x =4
Find the equation of the tangent at the given value of x.
\displaystyle
y = 2x^2 -7, x = -2
Find the equation of the tangent at the given value of x.
\displaystyle
f(x) = 3x^2 + 2x -5, x = -1
Find the equation of the tangent at the given value of x.
\displaystyle
f(x) = 5x^2 - 8x + 3, x = 1
Find the equation of the tangent to the graph of the function at the given value of x.
\displaystyle
f(x) = \frac{x}{x + 3}, x = -5
Find the equation of the tangent at the given value of x.
\displaystyle
y = \frac{2x + 5}{5x -1}, x = -1