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Hi parents and students! This is an introduction to what you’ll be needing to know by the end of the first Calculus semester. My courses cover all of the common core requirements (the syllabus for semester 1 is here, and semester 2 is here!) and prepare my students to think critically. Not only do I want your student to know the material in my class, I want them to succeed in all future classes and opportunities. Please try out this sample test to see what your child will be learning this semester!

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1. Determine the value of this limit.
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2. Determine the value of this limit.
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3. Determine the value of A so as to insure that this function is everywhere continuous.
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4a. Identify the value of this limit.
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4b. Identify the value of this limit.
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4c. Identify the value of this limit.
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4d. Identify the value of this expression.
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5. Find the instantaneous rate of change of f(x) = 2x² + 5x at x = 3 using the limit definition of the instantaneous derivative.
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6a. Using the formal definition of the derivative, find the slope of the tangent line to the curve given by f(x) = 1/(x – 6).
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6b. Find the slope of the normal line to the curve given by f(x) = 1/(x – 6).
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6c. Find the slope of the normal line to the curve given by f(x) = 1/(x – 6) at x = 0.
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7. Which of the following is (are) correct representation(s) of the derivative of f(x) evaluated at a?
All correct answers must be selected. No partial credit.
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8. Find f ‘ (θ) when f(θ) = cos(θ)cot(θ)
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9. Find g'(t) when g(t) = 3t/(t – 4)
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10a. Find the slope of the tangent line to the curve given by f(x) = (x³ + 7)(√2x)³
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10b. Find the equation of the tangent line (at x = 0) to the curve given by f(x) = (x³ + 7)(√2x)³
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11a. Using the functions f(x) and g(x) from the above graph, find f ’(2).
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11b. Using the functions f(x) and g(x) from the above graph, find g ’(2).
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11c. Using the functions f(x) and g(x) from the adjacent graph, find p ’(2) where p(x) = 3f(x)g(x).
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12. Find h ’(x) given h(x) = 9x³√x⁴ + 7
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13. Find f ’(x) where f(x) = sin( tan(x) ).
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14a. Find dy/dx when y = -4cos(11x).
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14b. Find d²y/dx² when y = -4cos(11x).
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## An object is moving rectilinearly in time with velocity given by v(t) = 6t – t² with units m/s. The position at t = 0 is 5 meters.

15a. Find the position function of time.
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15b. What are the appropriate units for 15a?
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15c. An object is moving rectilinearly in time with velocity given by v(t) = 6t – t² with units m/s. Assuming that position at t = 0 is 5 meters, find the acceleration function of time with appropriate units.
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15d. What are the appropriate units for 15c?
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16. Differentiate implicity to find y’ where 4x³y = 2x + sin(y)
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## Assume f(0) = 2, f'(0) = -1, f”(0) = 1, and g(x) = sec(3x)f(x). What is the equation of the line tangent to g(x) at x = 0.

17a. Identify the correct equation for g'(x).
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17b. What is the slope of the tangent line of g(x) at x = 0.
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17c. What is the equation of the tangent line of g(x) at x = 0.
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18a. Find g'(x) where g(x) = ( f(2x) )²
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18b. Use the data in this chart to find g'(3) where g(x) = ( f(2x) ) ²
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19. A large cube is losing volume at the rate of .2 m³/sec while still maintaining its cubical shape. What is the rate of change of the sides when the sides of the cube measure 2 meters?
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20. Consider the function f(x) = cos(x) + x over the interval [0, 1]radians. Show that the conditions of the Mean Value Theorem are met over this interval and then find the value of ξ predicted by the theorem.
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## For each x value below, state if it is a critical value or not.

21a. x = -7
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21b. x = -6
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21c. x = -5
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21d. x = -1
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21e. x = 1
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21f. x = 4
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21g. x = 7
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22. Which of the points in the graph of f(x) shown above have both a negative first derivative and also a negative second derivative value?
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23. A rectangular beam is cut out of a circular log of radius R. What is the maximum area of the log?
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24. What is the derivative of f(x) = ³√(4^(3x) + log₄x) ?
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25. Find f ’(x) where f(x) = arcsin(2x³/5).
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26. Find the value of the indefinite integral above.
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27. What is f(x) when f ’(x) = 1/cos²(x) and f(2π/3 radians) = 2?
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## Find f(x) when f ’’(x) = x + sin(5x) , f ’(π/6 radians) = 0, and f(0 radians) = 1.

28a. Find f ‘ (x) by taking the indefinite integral of f’’(x)
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28b. Find C₁
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28c. Find f(x) by taking the indefinite integral of f'(x)
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28d. Find C₂ and indentify the final formula for f(x).
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