2. Solve for x when x < 22. Write your answer in interval notation. 3. Find the distance between the points ( 1, 5) and (4, 3).

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1 Math 6 Practice Problems for Final. Find all real solutions x such that 7 3 x = 5 x 3.. Solve for x when 0 4 3x <. Write your answer in interval notation. 3. Find the distance between the points (, 5) and (4, 3). 4. Find all real solutions, x, such that x < 3x Find the equation for the line perpendicular to 3x 4y = and contains the point ( 6, ). 6. Given f(x) = x+ ( x + 3 and g(x) = x 4. Find the function and its domain. Write your answer in interval notation. f g ) (x) 7. Given f(x) = x 5 + x and g(x) = x 5 x + 3x 6, find (f g)(x) and evaulate (f g)(). 8. Given f(x) = (x + 3). Describe the transformations on the basic function (f(x) = x ). Then graph the function f(x) = (x + 3). Make sure to label some important points. 9. Graph the function 5 if x <, f(x) = 3x + if x < 3, x + 9 if x 3 0. Given f(x) = x + x and g(x) = e x, (f g)(x) = (g f)(0) =. Let h(x) = x + 5x 3. Find the x and y-intercepts.

2 . For h(x) = x +, find the average rate of change from x = to x =. 3. Find the inverse function of 3x+4 x. 4. Given the graph of a function below. Is that a graph of a one-to-one function? If yes, graph its inverse on the SAME graph. If not, explain why. y y = x x 5. Given f(x) = 4 (x + ) (x )(x 3). Find (a) all x-intercepts and y-intercept, (b) multiplicity of each zero of f (Indicate the multiplicity of which root), (c) leading term, (d) the graph the function. 6. Given f(x) = x. Find (a) all x-intercepts and y-intercept, (b)leading x +x term, (c)all asymptotes. (d) Graph the function (you may want to use the key number method to help). 7. Given f(x) = x +5x+6. Find (a) all x-intercepts and y-intercept, (b) x 3 leading term, (c)all asymptotes. (d) Graph the function (you may want to use the key number method to help).

3 8. Let f(x) = 3x 4 + 4x 3 x + 5 and g(x) = x x. Write f(x) = q(x)g(x) + r(x) where q(x) and r(x) are polynomials with deg r(x) < deg g(x). 9. List the potential rational zeros of the polynomial. f(x) = 3x 4 x 3 x Given f(x) = x 6 x 5 + 4x 3 0x + 3x + 5. If f(x) is divided by x +, what is the remainder? Is (x + ) a factor of f(x)?. Given f(x) = x + 3. Find (a) the domain, (b) the range, (c) the asymptote of the function. (d) Graph the function using the left grid and label at least three points.. Given f(x) = log (x ). Find (a) the domain, (b) the range, (c) the asymptote of the function. (d) Graph the function using the left grid and label at least three points. 3. Solve 5 x+ = Solve 7 3x = 3 x+4 and write your answer in terms of natural logarithm(i.e. ln). 5. Solve log 3 (x + ) + log 3 (x + 4) = 6. The number N of bacteria present in a culture at time t (in hours) obeys the law of uninhibited growth N(t) = 500e 0.0t. (a) Determine the number of bacteria at t = 0 hours. (b) What is the growth rate of the bacteria? (c) What is the population after 5 hours (d) When will the number of bacteria reach 3500? (e)when will the number of bacteria double? 7. Find an equation of the tangent line to the function f(x) = ln x at x = e. 3

4 8. Find the derivative of the following functions. (Do NOT simplify) (a) f(x) = 3x x 4x 3 x 5 (b) f(x) = (5x 4 x 3 5x)(e x x 3 + 6) (c) f(x) = 7 3x x 3 +4x x (d) f(x) = ln(6x + 3) (e) f(x) = x 3 e 7x (f) f(x) = e7x ln(x +4x) 9. Find the first and second derivative of the following function. (Do NOT simplify) f(x) = x 3 ln x 30. Given f(x) = (x + )(x ). (a) Find all x-intercepts and y-intercepts. (b) Locate all critical points. (c) Locate all local maximum and minimum. (d) Find intervals of increase and decrease. (e) Locate all points of inflection. (f) Find intervals on which the function is concave up and concave down. (g)draw the graph of f(x). LABEL (it means to WRITE down the words) local maximum and minimum and point of inflection. Give the coordinate pair of each important point (intercepts, local max and min and point of inflection) on the graph. 3. The total-cost, C(x), and total-revenue, R(x), functions for producing x items are shown below, where 0 x 500. C(x) = x and R(x) = x + 800x (a) Find the total-profit function P (x). (b) Find the marginal revenue and marginal cost. (c) Find the number of items, x, for which the total profit is a maximum. (d) Find the maximum total profit. 4

5 3. An open-top box is to be made by cutting small congruent squares from the corners of a -in.-by--in. sheet of tin and bending up the sides. (a) How large should the squares cut from the corners be to make the box hold as much as possible? (b) Find the maximum volume of the box. 33. Find an antiderivative of the function f(x) = 4x 3 + x. 34. Evaluate the integrals: (a) (x + )dx (b) (3x + x)dx (c) ( x + 3 x)dx (d) ( x + x 3 x )dx (e) (3e x + )dx x 35. Verify the integral formula by differentiation. (x + ) e x dx = (x + )e x + C 36. Evaluate the integrals: (a) (x 4 + 4x + ) (x 3 + x)dx (c) 3x +4x 6 dx x 3 +x 6x+ (b) 3x 7 3x dx (d) e x (e x + 0) 4 dx 37. Given the graph of f(x) in the following, find 7 f(x)dx y x 38. Suppose f(x)dx = 0, 4 f(x)dx = 3, f(x)dx, [3f(x) 5h(x)]dx, 4 f(x)dx. 4 h(x)dx =. Find 5

6 39. Estimate the following integral using (a) Left-endpoint Rule with n = 3, (b) Right-endpoint Rule with n = 3 and (c) Mid-point Rule with n = 3. 7 dx x 40. Find dy if y = x dx 0 + t dt. 4. Find dy dx if y = x 3 0 te t+ dt. 4. Evaluate the integrals: (a) 0 (x + 5)dx (b) (c) ( x + 3 x)dx (d) 0 (e) 3x x 3 + dx 0 (x + x)dx dx x 43. Find the total area of the region between the curve and the x-axis. y = x x, 3 x 44. Find the area of the region enclosed by the parabola y = x and the line y = x. 45. Find the volume of the solid generated by revolving the region bounded by y = x, y = 0 and x =. 46. The solid lies between planes perpendicular to the x-axis at x = 0 and x =. The cross sections perpendicular to the x-axis between these planes are squares with the one base point on the x-axis and one base point on the line y = x. Find the volume of the solid. 6

Comprehensive Practice Handout MATH 1325 entire semester

1 Comprehensive Practice Handout MATH 1325 entire semester Test 1 material Use the graph of f(x) below to answer the following 6 questions. 7 1. Find the value of lim x + f(x) 2. Find the value of lim