BC Calculus 12


Here you will find practice problems aligned with the Calculus 12 curriculum in the province of BC, Canada. Visit this site directly at hunkim.com/13

BC Calculus 12 Overview of Topics

Functions and Graphs

  • Parent functions from Pre-Calculus 12 (ex. exponential, logarithmic, polynomial, rational, trigonometric)
  • Piecewise functions
  • Inverse trigonometric functions
  1. Historical background: Calculus is the study of continuous c____________, and was developed independently in the late 17th century by N_________ and L___________.
    Solution
    change, Newton, Leibniz

  2. Calculus is the mathematical study of change.  Give an example of how Calculus is relevant to many fields of study such as Biology or Economics.
  3. Functions Review: Sketch:
    1. y=3^{-x}-2
    2. y=e^x+2
    3. y=\log_2(x-1)
    4. y=\frac{-\ln x^3}{2}
    5. y=x^3-3x^2+4
    6. \frac{3x-2}{x+1}
    7. \frac{2x^2-4x}{x^3-x^2-2x}
    8. \tan \left(\frac{2\pi x}{3}\right)
    9. Sketch the piecewise function below:
      \begin{cases} x^2-2x+1 & x<4 \\ 6 & x=4 \\ \frac{x}{2}+7 & x>4 \end{cases}
    10. y=\arcsin x
    11. y=\cos^{-1} x
    12. f(\theta)=\arctan \theta

Limits and Continuity

  1. Polynomials are smooth and c____________.
  2. Is the function f(x)=|x-2| continuous at x=2?
  3. What is the formal definition of a limit?
  4. A function is said to be continuous on the interval [a,b] if \lim_{x\to a} f(x)=f(a). If f(x) is continuous at x=a then, \lim_{x\to a^{-}} f(x)=?
  5. |x-2|<5
    1. Solve x expressing the domain as an inequality statement
    2. State the domain in plain English
    3. Now explain the meaning of |x-a|<\delta in plain English
  6. Solve |x+3|<4
  7. Find \lim_{x\to1}(5x-3)
  8. Find \lim_{x\to2} \frac{x-2}{x^2-4}
  9. f(x)=\lfloor x\rfloor
    1. Find \lim_{x\to2^+}f(x)
    2. Find \lim_{x\to2^-}f(x)
    3. Find \lim_{x\to2}f(x)
  10. f(x)=\frac{1}{x}
    1. Find \lim_{x\to0}f(x)
    2. Find \lim_{x\to0^-}f(x)
  11. See f(x) below:
    1. Evaluate f(4)
    2. Evaluate \lim_{x\to4}f(x)
  12. f(x) is defined below:
    \begin{cases} (x-2)^2 & x>3 \\ (x+k) & x<3 \\ 2 & x=3 \end{cases}
    1. Find k so that \lim_{x\to3^-} f(x)=\lim_{x\to3^+} f(x)
    2. If possible, find f(3)
    3. If possible, find \lim_{x\to3} f(x)
  13. If possible, find \lim_{x\to2^-} in the graph below:
  14. Find \lim_{x\to2^-} \frac{1}{x^2-4}
  15. Give an example of:
    1. Jump discontinuity
    2. Infinite discontinuity
    3. Oscillating discontinuity
  16. Find \lim_{x\to1} \frac{2(x-1)}{|x-1|}
  17. Find \lim_{x\to\infty} \left(-3^{-x}+2\right)
  18. Temperature K(t)=100\left(\frac{1}{2}\right)^{0.1t}+20
    1. Initial temperature?
    2. Find \lim_{t\to\infty}K(t)
  19. Use the Intermediate Value Theorem to show that a root exists between x=1 and x=3 for P(x)=x^3-3x+2
  20. \lim_{x\to0} x^2\sin\left(\frac{1}{x}\right)
  21. Enrichment: Prove using geometry and algebra that \lim_{x\to0}\frac{\sin x}{x}
  22. Find \lim_{x\to0}\frac{1-\cos x}{x}
  23. Find \lim_{x\to\infty}\frac{2x-4}{x+1}
  24. Find \lim_{x\to\infty}\frac{6x^2-4x}{4x^5-2x^3+x-3}
  25. Find \lim_{x\to\infty} \frac{4x^3-2x^2+x}{6x^3+3x^2-5}
  26. Find \lim_{x\to0} \frac{\sin5x}{3x}
  27. Find \lim_{x\to7} \frac{\sqrt{x+2}-3}{x-7}

Differentiation

  • Power rule, product rule, quotient rule, and chain rule
  • Transcendental functions:  logarithmic, exponential, trigonometric
  • Higher order, implicit
  1. Given y=x^2, find the average slope from:
    1. x=0 to x=3
    2. x=2 to x=3
    3. x=2.9 to x=3
    4. Can you guess the instantaneous slope at x=3?
  2. We can easily find the slope formula by using the power rule:
    Given y=x^n the slope formula is y'=m=nx^{n-1}. Find y' the slope formula when:
    1. y=x^3
    2. y=2x^2-5x
    3. y=5x^4
    4. y=7x
    5. y=4
    6. What is your best time in using the Power Rule at thatquiz.org?
    7. Prove this as an enrichment challenge.
  3. What is the domain in which polynomials are continuous and differentiable?
  4. f(x)=\sqrt{9-x^2}
    1. Sketch
    2. What is the domain in which f(x) is differentiable?
    3. What is the domain in which f(x) is continuous?
  5. f(x)=|x|
    1. Is this function continuous at x=0?
    2. Is this function differentiable at x=0?
  6. f(x)=x^{2/3}
    1. Is this function continuous at x=0?
    2. Is this function differentiable at x=0?
  7. f(x)=\frac{x-1}{|x-1|}
    1. Is this function continuous at x=1?
    1. Is this function differentiable at x=1?
  8. Given y=x^4, find:
    1. y'
    2. y''
    3. y'''
    4. y^{(4)}
  9. Give f(x)=x^3-2x^2 evaluate f'(1)
  10. See diagram below:

    Explain the limit definition of slope f'(a)=\lim_{h\to 0} \frac{f(a+h)-f(a)}{h}
  11. Given f(x)=x^2, find f'(3)
    1. Using this limit definition of slope
    2. Using the power rule
  12. See diagram below:
    1. Use this diagram to explain the alternate limit definition of slope: f'(x)=\lim_{x\to a} \frac{f(x)-f(a)}{x-a}
    2. Now solve the previous question f'(3) using this alternate definition
  13. Use the limit definition of slope to find f'(x) given:
    1. f(x)=\frac{3}{x}
    2. f(x)=\sqrt{x}
  14. Interpret the meaning of: \lim_{x\to0}\frac{(x+h)^2-9}{h}
    Solution
    f(x)=x^2, f'(3)
  15. f(x)=x^4+2x
    1. Find the equation of the tangent line at x=1
    2. Find the equation of the normal line at x=1
  16. Enrichment: use the limit defintion of slope to prove that the derivative of \sin x is \cos x
  17. Given f(x)=\sin x find:
    1. \frac{d}{dx} f(x)
    2. \frac{dy}{dx}
    3. f''(x)
    4. \frac{d^3y}{dx^3}
    5. f^{(123)}(\pi/3)
  18. Enrichment – Show that:
    1. \frac{d}{dx} e^x=e^x
    2. \frac{d}{d\theta} \left(\sin^{-1} \theta\right)=\frac{1}{\sqrt{1-\theta^2}}
  19. Try to memorize a common derivative formula sheet
  20. Kinematics – Position is a function of time:
    s(t)=-t^3-2t^2+3t-4
    1. What is the meaning of \frac{ds}{dt} or s'(t)?
    2. Find the speed of the particle at t=1
    3. What is the meaning of \frac{dv}{dt} or v'(t)?
    4. Find the initial acceleration of the particle.
  21. Find the derivative:
    1. \frac{d}{dx}3
    2. \frac{d}{dx}5x
    3. \frac{d}{dx}x^{-2}
    4. \frac{d}{dx}x^{-1}
    5. \frac{d}{d\theta}\tan \theta
    6. \frac{d}{dx}\csc x
    7. \frac{d}{dx}7^x
    8. \frac{d}{dx}\cos^{-1}x
    9. \frac{d}{dx}\log_2 x
  22. Enrichment: Where doe the Product Rule come from?
    \left( fg\right)'=f'g+g'f
  23. State the Quotient Rule
  24. Differentiate:
    1. y=\sqrt[3]{x^2}(2x-x^2)
      1. By expanding first
      2. By using the product rule
    2. \frac{x^2}{x^{-5}}
      1. By using exponent laws
      2. By using the quotient rule
      3. By rewriting the question as a product rule problem
    3. W(z)=\frac{3z+9}{2-z}
  25. Suppose that the amount of air in a balloon at any time t is given by V(t)=\frac{6\sqrt[3]{t}}{4t+1}. Determine if the balloon is being filled or drained of air at t=8.
  26. Use the quotient rule to prove that the derivative of \tan A is \sec^2A.
  27. Differentiate:
    1. y=\sin x(x^2)
    2. y=\frac{\log_2 x}{e^x}
    3. y=2x \arcsin(x)
  28. Enrichment: Justify the Chain Rule
  29. \frac{dz}{dx}=\frac{dz}{dy}\times ?
  30. Given F(x)=f\left( g(x)\right), find F'(x)
  31. Differentiate:
    1. f(x)=\sin(3x^2+x)
    2. y=\sec(1-5x)
    3. y=\tan^2 {2x+e^x}
    4. y=\frac{\left(x^3+4\right)^5}{\left(1-2x^2\right)^3}
    5. y=\frac{2^{3x}}{\cos(2x)-x}
    6. F(x)=\tan^{-1}(3x)\sqrt[3]{1-2x^2}
  32. Find \frac{dy}{dx} given y=x^x
  33. Challenge: Given y=(x^x)^x find the equation of the normal line at x=1
  34. Use Implicit Differentation to find \frac{dy}{dx}
    1. x^2+y^2=9
    2. x^3y^5+2x=8y^3+1
    3. x^3\tan(y)+y^5\sec(x)=3x
    4. e^{2x+5y}=x^3-\ln(xy^2)
  35. Now use explicit differentiaion to find \frac{dy}{dx} given x^2+y^2=9
  36. What is the equation of the tangent line of a circle with radius 5 (centered at the origin) that goes through the point (3,4)?
  37. Enrichment: Use implicit differentiation to prove:
    1. \frac{d}{dx} \sin^{-1}x=\frac{1}{\sqrt{1-x^2}}
    2. \frac{d}{dx} \sin^{-1}\left(\frac{x}{k}\right)=\frac{1}{\sqrt{k^2-x^2}}

Applications of Differentiation

  • Relating graph of f(x) to f'(x) to f''(x)
  • Increasing / decreasing, concavity
  • Differentiability, mean value theorem
  • Newton’s method
  • Problems in contextual situation including:
    • Related rates
    • Optimization
  1. P(x)=x^3-3x+2
    1. Find the P intercept
    2. Sketch by using the Factor Theorem
    3. Now sketch by finding the critical points when f'(x)=0
    4. Now sketch by finding the critical points when f'(x)=0
  2. f(x)=(x-2)^2(x^2+2x+1)
    1. Sketch using Calculus techniques (ex. f'(x)=0
    2. Find the coordinates of the local / relative minimum within the interval x\in[0,3]
    3. Find the coordinates of the global / absolute maximum within the interval x\in[0,3]
  3. Use Calculus to sketch
    1. f(x)=\frac{1}{x^2-9}
    2. h(t)=t(4-t)^{2/3}
    3. f(x)=\frac{1}{\sqrt{9-x^2}}
  4. Mean Value Theorem
    1. Determine the c values which satisfy the conclusions of the Mean Value Theorem for f(x)=x^3+2x^2-x on [-1,2]
    2. f(x)=|x|/x. Find the average slope from x=-1 to x=1. Explain why the Mean Value Theorem does not fail.
  5. Newton’s Method:
    1. Enrichment: How was Newton’s Method derived? x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}
    2. Use Newton’s method to solve x^3+3x+1=0 accurate to 5 decimal places. Use x_1=-0.3 as your first approximation
    3. Sketch an example of how Newton’s Method can fail
    4. Use Newton’s method to determine an approximation to the solution \cos x=x that lies in the interval [0,2] correct to six decimal places.
    5. Solve (x-2)^{1/3}=0 using Newton’s Method using x=3 as your initial guess.
  6. Optimization:
    1. We need to enclose a rectangular field with a fence. We have 400 feet of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.
    2. We want to construct a box whose base length is 4 times the base width. The material used to build the top and bottom cost $20/ft2 and the material used to build the sides cost $10/ft2. If the box must have a volume of 100 ft3 determine the dimensions that will minimize the cost to build the box.
    3. We want to construct a box with a square base and we only have 20 m2 of material to use in construction of the box. Assuming that all the material is used in the construction process determine the maximum volume that the box can have.
    4. A manufacturer needs to make a cylindrical can that will hold 4 liters of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction.
    5. We have a piece of cardboard that is 12 inches by 8 inches and we’re going to cut out the corners (making equal sized squares in each corner) and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.
    6. A printer need to make a poster that will have a total area of 250 in2 and will have 1 inch margins on the sides, a 2 inch margin on the top and a 3 inch margin on the bottom. What dimensions will give the largest printed area?
  7. Related Rates:
    1. See diagram below:

      A balloon is rising vertically at point P. Your horizontal distance to the balloon is 400 ft.
      1. How fast is the balloon rising at the moment \theta=\frac{\pi}{3} radians?
      2. Find \frac{d\theta}{dt} at this moment
    2. See diagram below:

      A police car at A is heading south in pursuit of your car at B which is heading east. When the police car is 0.6 km north of the intersection and you are 0.8 km to the east, the diagonal distance between the police and you is increasing at 10 kph. If the police is moving at 120 kph at this moment in time, how fast are you travelling?
    3. See diagram below:

      You fill a cone with maple syrup at the rate of 6 cubic meters / min. The height of the cone is 10 m and the radius of the circle is 4 m. How fast is the liquid level rising when the syrup is 8 meters deep?

Test yourself here. You may use technology but show your work as if you had no calculator.

Integration

  • Definition of an integral
  • Notation
  • Definite and indefinite
  • Approximations
    • Riemann sum, rectangle approximation method, trapezoidal method
  • Fundamental theorem of calculus
  • Methods of integration
    • Antiderivatives of functions
    • Integration by substitution
    • Integration by parts
  1. f(x)=x^2. Estimate the area under the curve from x=0 to x=6 using 3 subinterval Riemann sum while using:
    1. LRAM
    2. RRAM
    3. MRAM
    4. Trapezoidal method
      1. Estimate the area under the curve
      2. Enrichment: where does the trapezoid area formula come from?
        A_{\text{trapezoid}}=\frac{a+b}{2}h
  2. \int_2^5 t^2\text{ }dt. Identify:
    1. The upper limit of integration
    2. The Integral sign
    3. The lower limit of integration
    4. The integrand
    5. The variable of integration
    6. How is this expression read?
    7. Now evaluate this integral
  3. Now find the exact area under the curve f(x)=x^2 from x=0 to x=6
  4. Indefinite integrals:
    1. \int x^2\,dx
    2. \int \cos x\,dx
    3. \int 4x^3\,dx
    4. \int 5\,dx
    5. \int e^x\,dx
    6. \int \sec x \tan x\,dx
  5. For indefinite integrals, why do we need to remember to add the +c constant?
  6. Applying FTC Part I
    1. \frac{d}{dx}\int_0^x t^2\,dt
    2. Find \frac{dy}{dx} if y=\int_1^{x^3} \sin{t}\,dt
  7. Applying FTC Part II: Definite integrals
    1. \int_0^3 x^2\,dx
    2. \int_0^{\frac{\pi}{2}} \cos x\,dx
  8. Integration by substitution:
    1. \int \cos(2x)\,dx
    2. \int (x+1)^4\,dx
    3. \int \sqrt{4x-1}\,dx
    4. \int (5x-2)\,dx
    5. \int \left(2x^3+1\right)^7(x^2)\,dx
    6. \int \frac{1}{\cos^2 3x}\,dx
    7. \int \sin^3 x \cos x\,dx
    8. \int (x^2+2x-3)(x+1)\,dx
    9. \int_0^{\pi/3} \tan x \sec^2 x\,dx
    10. Challenge: \int \tan x\,dx
  9. Enrichment:
    1. Explain FTC Part I
    2. Explain FTC Part II
  10. Integration by parts:
    1. Enrichment: Given y=uv, use implicit differentiation to show that: \int u\,dv=uv-\int v\,du
    2. \int x \cos x\,dx
    3. What is the acronym LIPET used for?
    4. Find the area of the region bounded by the curve y=xe^-x at the x-axis from x=0 to x=2
    5. Repeated integration by parts: \int x^2 e^x\,dx
    6. Clever algebra challenge: \int e^x \cos x\,dx

Applications of Integration

  • area under a curve, volume of solids, average value of functions
  • differential equations
  • initial value problems
  • slope fields
  1. Evaluate \int_{-2}^2 2x+3\,dx
    1. Using FTC2
    2. Using basic geometric shapes
  2. Find \int_{-3}^3 \tan x\,dx using your knowledge of odd functions
  3. Find the area under the curve:
    1. f(x)=x^2 betwen x=0 to x=4
    1. y=\cos \theta, 0\leq\theta\leq \frac{\pi}{2}
    2. h(t)=\frac{1}{t}, t\in[1,3]
  4. Find the area between enclosed by y=\sqrt{x}, y=x-2 and the x-axis.
    1. By subtracting a common geometrical shape
    2. By combining definite integrals
    3. By integrating with respect to y instead of x
  5. \int_{-1}^2 x^3\,dx
  6. Find the area between the graphs:
    1. f(x)=1 and g(x)=\cos^2 x
    2. y=x(x-2)^2 and y=3x-6
    3. y_1=3x-x^2 and y_2=2x^3-x^2-5x
  7. Average value of a function:
    1. Explain the average value of a function formula:
      av(f)=\frac{1}{b-1} \int_a^b f(x)\,dx
    2. f(x)=9x^2 on the interval x\in[0,2]
      1. Find the average value of this function
      2. Find the c value in which this average value occurs within the given interval
    3. f(x)=x^2-1 on the interval x\in[0,\sqrt{8}]
      1. Find the average value of this function
      2. Find the c value in which this average value occurs within the given interval
  8. Differential equations:
    1. An equation that contains a derivative like \frac{dy}{dx}=y \ln x is a d_________ equation
    2. Solve the initial value problem:
      1. Find the general solution of \frac{dy}{dx}=\frac{1}{x}+5
        Solution
        y=ln|x|+5x+C
      2. \frac{dy}{dx}=\sin x, Initial condition: y(0)=2
      3. \frac{dy}{dx}=\tan x, Initial condition: y(3)=5
      4. Recall that acceleration a=-9.8. You throw a ball straight up. You release the ball from a height of 2 m with a speed of 10 m/sec.
        1. Find the position and velocity functions
        2. When does the ball hit the ground?
        3. How fast is the ball travelling at this time?
          Solution
          a. \frac{dv}{dt}=-9.8 and v(0)=10
          v=-9.8t+C
          10=-9.8(0)+C
          10=C
          v=-9.8t+10
          \frac{ds}{dt}=-9.8t+10, s(0)=2
          s=-9.8\left(\frac{1}{2}t^2\right)+10t+K
          2=-4.9(0)^2+10(0)+K
          2=K
          s=-4.9t^2+10t+2
          0=-4.9t^2+10t+2
          Using the quadratic formula:
          t\approx 2.2243 (reject t=-0.1835)
          v=-9.8(2.2243)+10\approx -11.7983 so the ball is heading down at the speed of 11.8 m/sec
    3. $1000 is invest in an investment account that pays 8% interest compounded continuously.
      1. Find a formula for the amount of money in the account. Hint: \frac{dy}{dt}=0.08y, Initial condition: y(0)=1000
        Solution

        y(t)=Ce^{0.08t} is a solution to the differential equation for any real number C because:
        \frac{dy}{dy}=C(0.08e^{0.08t}=0.08\left(Ce^{0.08t}\right)=0.08y(t)
        Applying the initial condition we find:
        y(t)=Ce^{0.08t}
        y(0)=Ce^{(0.08)(0)}
        1000=C
        Thus y(t)=1000e^{0.08t}
      2. How much will your investment grow to be in 50 years if you stop making contributions?
        Solution
        y(t)=1000e^{0.08t}
        y(50)=1000e^{(0.08)(50)}\approx \$54 598
  9. Volume of Solids:
    1. See diagram below:

      This square based pyramid has a maximum side length of 2 m on each side. Each cross section of the pyramid parallel to the base is a square of side length x. Find the volume of the pyramid.
      Solution
      V=\int_0^2 A(x)\,dx=\int_0^3 x^2=\frac{8}{3}\text{ m}^3
    2. The region between the graph of f(x)=3+x \cos x and the x-axis over the interval [-3,3] is revolved about the x-axis to generate a solid. Use technology to find the volume of the solid.
      Solution
      A(x)=\pi\left(f(x)\right)^2
      The volume of the solid is:
      V=\int_{-3}^{3}A(x)\,dx
      V=\int_{-3}^{3} \pi(3+x \cos x)^2\,dx\approx 63.25
    3. Below are the graphs y=\sin x and y=\cos x. The dotted region is revolved about the x-axis to form a solid. Find its volume

      Solution
      V=\int_0^{\pi/4} \pi R^2-\pi r^2
      V=\int_0^{\pi/4} \pi(\cos^2 x-\sin^2 x)\,dx