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School City of Hobart Calculus In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives - operating computer equipment, planning timelines and schedules, reading and interpreting data, comparing prices, managing personal finances, and completing other problem-solving tasks. What they learn in mathematics and how they learn it will provide an excellent preparation for a challenging and ever-changing future. |
| Calculus and Pre-Calculus |
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Limits and Continuity
The learner will be able to understand the concept of limit, find limits of functions at points and at infinity, decide if a function is continuous, and use continuity theorems.
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C.1.1
The learner will be able to understand the concept of limit and estimate limits from graphs and tables of values.
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C.1.2
The learner will be able to find limits by substitution.
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C.1.3
The learner will be able to find limits of sums, differences, products, and quotients.
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C.1.4
The learner will be able to find limits of rational functions that are undefined at a point.
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C.1.5
The learner will be able to find one-sided limits.
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C.1.6
The learner will be able to find limits at infinity.
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C.1.7
The learner will be able to decide when a limit is infinite and use limits involving infinity to describe asymptotic behavior.
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C.1.8
The learner will be able to find special limits. Example: Use a diagram to show that the limit above is equal to 1.
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C.1.9
The learner will be able to understand continuity in terms of limits.
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C.1.10
The learner will be able to decide if a function is continuous at a point.
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C.1.11
The learner will be able to find the types of discontinuities of a function.
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C.1.12
The learner will be able to understand and use the Intermediate Value Theorem on a function over a closed interval.
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C.1.13
The learner will be able to understand and apply the Extreme Value Theorem: If f(x) is continuous over a closed interval, then f has a maximum and a minimum on the interval.
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Differential Calculus
The learner will be able to find derivatives of algebraic, trigonometric, logarithmic, and exponential functions. They find derivatives of sums, products, and quotients, and composite and inverse functions. They find derivatives of higher order, and use logarithmic differentiation and the Mean Value Theorem.
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C.2.1
The learner will be able to understand the concept of derivative geometrically, numerically, and analytically, and interpret the derivative as a rate of change.
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C.2.2
The learner will be able to state, understand, and apply the definition of derivative.
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C.2.3
The learner will be able to find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.
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C.2.4
The learner will be able to find the derivatives of sums, products, and quotients. Example: Find the derivative of x cos x. .
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C.2.5
The learner will be able to find the derivatives of composite functions, using the chain rule.
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C.2.6
The learner will be able to find the derivatives of implicitly-defined functions.
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C.2.7
The learner will be able to find derivatives of inverse functions.
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C.2.8
The learner will be able to find second derivatives and derivatives of higher order.
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C.2.9
The learner will be able to find derivatives using logarithmic differentiation.
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C.2.10
The learner will be able to understand and use the relationship between differentiability and continuity.
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C.2.11
The learner will be able to understand and apply the Mean Value Theorem.
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Applications of Derivatives
The learner will be able to find slopes and tangents, maximum and minimum points, and points of inflection. They solve optimization problems and find rates of change.
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C.3.1
The learner will be able to find the slope of a curve at a point, including points at which there are vertical tangents and no tangents.
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C.3.2
The learner will be able to find a tangent line to a curve at a point and a local linear approximation. Example: In the last example, find an equation of the tangent at (2, 8). .
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C.3.3
The learner will be able to decide where functions are decreasing and increasing. Understand the relationship between the increasing and decreasing behavior of f and the sign of f'.
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C.3.4
The learner will be able to find local and absolute maximum and minimum points. Example: In the last example, find the local maximum and minimum points of f(x). .
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C.3.5
The learner will be able to analyze curves, including the notions of monotonicity and concavity. Example: In the last example, for which values of x is f(x) decreasing and for which values of x is f(x) concave upward? .
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C.3.6
The learner will be able to find points of inflection of functions. Understand the relationship between the concavity of f and the sign of f". Understand points of inflection as places where concavity changes. Example: In the last example, find the points of inflection of f(x) and where f(x) is concave upward and concave downward. .
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C.3.7
The learner will be able to use first and second derivatives to help sketch graphs. Compare the corresponding characteristics of the graphs of f, f', and f". Example: Use the last examples to draw the graph of f(x) = x3 - 3x. .
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C.3.8
The learner will be able to use implicit differentiation to find the derivative of an inverse function. .
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C.3.9
The learner will be able to solve optimization problems.
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C.3.10
The learner will be able to find average and instantaneous rates of change. Understand the instantaneous rate of change as the limit of the average rate of change. Interpret a derivative as a rate of change in applications, including velocity, speed, and acceleration.
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C.3.11
The learner will be able to find the velocity and acceleration of a particle moving in a straight line.
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C.3.12
The learner will be able to model rates of change, including related rates problems. Example: A boat is heading south at 10 mph. Another boat is heading west at 15 mph toward the same point. At these speeds, they will collide. Find the rate that the distance between them is decreasing 1 hour before they collide.
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Integral Calculus
The learner will be able to define integrals using Riemann Sums, use the Fundamental Theorem of Calculus to find integrals, and use basic properties of integrals. They integrate by substitution and find approximate integrals.
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C.4.1
The learner will be able to use rectangle approximations to find approximate values of integrals.
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C.4.2
The learner will be able to calculate the values of Riemann Sums over equal subdivisions using left, right, and midpoint evaluation points.
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C.4.4
The learner will be able to understand the Fundamental Theorem of Calculus: Interpret a definite integral of the rate of change of a quantity over an interval as the change of the quantity over the interval, that is f'(x)dx = f(b) - f(a).
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C.4.3
The learner will be able to interpret a definite integral as a limit of Riemann Sums. Example: Find the values of the Riemann Sums over the interval [0, 3] using 12, 24, etc., subintervals of equal width for f(x) = x2 evaluated at the midpoint of each subinterval. Find the limit of the Riemann Sums. .
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C.4.5
The learner will be able to use the Fundamental Theorem of Calculus to evaluate definite and indefinite integrals and to represent particular antiderivatives. Perform analytical and graphical analysis of functions so defined.
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C.4.6
The learner will be able to understand and use these properties of definite integrals.
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C.4.7
The learner will be able to understand and use integration by substitution (or change of variable) to find values of integrals.
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C.4.8
The learner will be able to understand and use Riemann Sums, the Trapezoidal Rule, and technology to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
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Applications of Integration
The learner will be able to find velocity functions and position functions from their derivatives, solve separable differential equations, and use definite integrals to find areas and volumes.
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C.5.1
The learner will be able to find specific antiderivatives using initial conditions, including finding velocity functions from acceleration functions, finding position functions from velocity functions, and applications to motion along a line. Example: A bead on a wire moves so that its velocity, after t seconds, is given by v(t) = 3 cos 3t. Given that it starts 2 cm to the left of the midpoint of the wire, find its position after 5 seconds. .
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C.5.2
The learner will be able to solve separable differential equations and use them in modeling.
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C.5.3
The learner will be able to solve differential equations of the form y' = ky as applied to growth and decay problems. Example: The amount of a certain radioactive material was 10 kg a year ago. The amount is now 9 kg. When will it be reduced to 1 kg? Explain your answer. .
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C.5.4
The learner will be able to use definite integrals to find the area between a curve and the x-axis, or between two curves.
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C.5.5
The learner will be able to use definite integrals to find the average value of a function over a closed interval.
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C.5.6
The learner will be able to use definite integrals to find the volume of a solid with known cross-sectional area. Example: A cone with its vertex at the origin lies symmetrically along the x-axis. The base of the cone is at x = 5 and the base radius is 7. Use integration to find the volume of the cone. .
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C.5.7
The learner will be able to apply integration to model and solve problems in physics, biology, economics, etc., using the integral as a rate of change to give accumulated change and using the method of setting up an approximating Riemann Sum and representing its limit as a definite integral. Example: Find the amount of work done by a variable force. .
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