Introduction to Calculus

Introduction to Calculus Harmanpreet, Richelle, Umar

Question 1 Rationalize the denominator of:

Question 2 Find the slope of the tangent to at x=9

Question 3 Determine the slope of the tangent to each curve at the given point.

Question 4 Find the slope of PQ, in simplified form, given P(1, -1) and Q(1+h), f(1+h)), where f(x)=-x2

Question 5 The function s(t)=6t(t+1) describes the distance (in kilometres) that a car has travelled after a time t (in hours) for • Calculate average velocity of the car from t=2 to t=3 • Find the instantaneous velocity of the car at t=2

Question 6 Evaluate:

Question 7 • Evaluate the limit of the indeterminate quotient:

Question 8 Evaluate the limit:

Question 9 The height, in metres, of an object that has fallen from a height of 180 m is given by the position function s(t) = -5t2 +180, where t≥0 and t is in seconds. • Find the average velocity during the first 2 seconds • Find the velocity of the object when t=4 • At what velocity will the object hit the ground?

Question 10 Explain why the given limit does not exist:

Question 11 Consider the graph of the function f(x)= 5x2-8x. Calculate the slop of the secant that joins the points on the graph given by x=-2 and x=1

Question 12 Determine the slope of the tangent at x=4 for

Question 13 Rationalize the numerator of each of the following expressions:

Question 14 Determine the slope of the tangent to the graph of the parabola f(x)=x2 at P(3, 9)

Question 15 A toy rocket is launched straight up so that its height s, in metres, at time t, in seconds, is given by s(t)= -5t2 + 30t + 2. What is the velocity of the rocket at t=4?

Question 16 Sketch a graph of the following function, and determine if it is continuous.

Question 17 Sketch the graph of any function that satisfies the given conditions:

Question 18 If use the properties of limits to evaluate the limit:

Question 19 Evaluate, if the limit exists:

Question 20 A weather balloon is rising vertically. After t hours, its distance above the ground, measured in kilometres, is given by the formula S(t)= 8t-t2. • Determine the average velocity of the weather balloon from t=2 h to t=5 h. • Determine its velocity at t=3 h

Question 21 • Evaluate the limit:

Question 22 Rationalize the denominator.

Question 23 Determine the slope of the tangent to the rational function at point (2, 6).

Question 24 • A pebble is dropped from a cliff, 80 m high. After t seconds, the pebble is smetres above the ground, where s(t) = 80 – 5t2, 0 t 4. • Calculate the average velocity of the pebble between the times t = 1 second and t = 2 seconds • Calculate the average velocity of the pebble between the times t = 1 second and t = 1.5 seconds. • Explain why your answers for parts a and b are different.

Question 25 Suppose that the temperature T, in degrees Celsius, varies with the height h, in kilometers, above Earth’s surface according to the equation . Find the rate of change in temperature with respect to a height of 3 km.

Question 26 Find equation of the tangent at the given value of x. f(x) = 5x2 – 8x + 3 , x = 1

Question 27 Evaluate.

Question 28 Evaluate

Question 29 Evaluate, if the limit exists

Question 30 a. Sketch the graph of the following function:f(x) = x + 1, if x < -1 - x + 1, if - 1 x < 1 x – 2, if x > 1 b. Find all values at which the function is discontinuous.c. Find the limits at those values, if they exist

Question 31 The estimated population of a bacteria colony is P(t) = 20 + 61t + 3t2, where the population P, is measured in thousands at t hours.a. What is the estimated population of the colony at 8 hours?b. What rate is the population changing at 8 hours?

Question 32 Determine the constants a and b such that f(x) is continuous for all values of x. F(x) = ax + 3, if x > 5 8, if x = 5 x2 + bx + a, if x < 5

Question 33 Examine the continuity of g(x)= x+3 when x=2

Introduction to Calculus