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Quizzes > High School Quizzes > Mathematics

AP Calculus AB Unit 1 Practice Test

Master Key Concepts with Focused Practice Quizzes

Difficulty: Moderate
Grade: Grade 12
Study OutcomesCheat Sheet
Colorful paper art promoting a Calculus AB Kickoff quiz for high school students.

What does the derivative of a function at a point represent?
The slope of the tangent line to the function at that point
The average rate of change of the function over an interval
The function's y-intercept at that point
The area under the function's curve at that point
The derivative represents the instantaneous rate of change; it is geometrically the slope of the tangent line at that point. Average rate of change, y-intercept, and area are not part of the definition of the derivative.
Using the power rule, what is the derivative of f(x) = x^3?
3x^2
x^2
3x^3
x^3
The power rule states that d/dx(x^n) = n*x^(n-1). For x^3, the derivative is 3x^2. The other options either omit the multiplication factor or simply restate the original function.
What is the derivative of a constant function?
0
The constant itself
Undefined
1
The derivative of any constant function is 0 because there is no change in the function's value as x varies. This is a direct consequence of the definition of the derivative.
Which of the following represents the limit definition of the derivative f'(a)?
lim (h→0) [f(a+h) - f(a)]/h
lim (h→∞) [f(a+h) - f(a)]/h
lim (h→0) [f(a+h) - f(a-h)]/(2h)
lim (h→0) [f(a+h) + f(a)]/h
The correct limit definition of the derivative f'(a) is given by lim (h→0) [f(a+h) - f(a)]/h, which calculates the instantaneous rate of change. The other options use incorrect limits or operations that do not yield the derivative.
What property must a function have at a point to be differentiable there?
It must be continuous at that point
It must have a defined maximum at that point
It must be symmetric about that point
It must be integrable at that point
For a function to be differentiable at a point, it must be continuous there. While other properties might be present, continuity is a necessary condition for differentiability.
What is the derivative of f(x) = x^2 sin(x) using the product rule?
2x sin(x) + x^2 cos(x)
2x cos(x) + x^2 sin(x)
sin(x) + cos(x)
2x sin(x) - x^2 cos(x)
Using the product rule, where the derivative of a product is f'·g + f·g', we differentiate x^2 to get 2x and sin(x) to get cos(x). Therefore, the derivative is 2x sin(x) + x^2 cos(x).
Which of the following functions is continuous on all real numbers?
f(x) = x^3 - 2x + 5
f(x) = 1/(x - 1)
f(x) = √x
f(x) = 1/x^2
Polynomials, like x^3 - 2x + 5, are continuous for all real numbers because they have no breaks, holes, or vertical asymptotes. The other functions have points in their domain where continuity fails.
If f(x) = ln(x), what is its derivative?
1/x
ln(x)
x
1/(x ln(x))
The derivative of the natural logarithm function ln(x) is 1/x, valid for x > 0. This follows from standard differentiation rules for logarithmic functions.
Solve: lim (x→2) (x² - 4)/(x - 2).
4
0
Undefined
2
Factor the numerator as (x - 2)(x + 2) and cancel the common term with the denominator, leaving x + 2. Substituting x = 2 yields 4. The other options do not follow from the proper simplification process.
Let F(x) = ∫₝ˣ f(t) dt. According to the Fundamental Theorem of Calculus, what is F'(x)?
f(x)
F(x)
f(a)
f'(x)
The Fundamental Theorem of Calculus states that if f is continuous on an interval, then the derivative of F(x) = ∫₝ˣ f(t) dt is f(x). The other answers confuse the roles of the function and its integral.
Which rule is used to differentiate a composite function?
Chain Rule
Product Rule
Quotient Rule
Sum Rule
The Chain Rule is specifically used for differentiating composite functions, allowing one to take the derivative of the outer function and multiply it by the derivative of the inner function.
What is the derivative of f(x) = e^(3x)?
3e^(3x)
e^(3x)
3e^(x)
e^(x)
When differentiating e^(3x), the chain rule requires multiplying by the derivative of the exponent, which is 3. Thus, the derivative is 3e^(3x).
What is the derivative of sin(x)?
cos(x)
sin(x)
-cos(x)
-sin(x)
The standard differentiation rule for trigonometric functions states that the derivative of sin(x) is cos(x). This is a foundational result in calculus.
Using the quotient rule, find the derivative of f(x) = (2x³ - x)/x².
2 + 1/x²
2 - 1/x²
2x
2x - 1/x²
By simplifying f(x) to 2x - 1/x and then differentiating term-by-term, the derivative is found to be 2 + 1/x². Alternatively, applying the quotient rule directly yields the same result after simplification.
Find the antiderivative of 3x² dx.
x³ + C
3x² + C
x² + C
3x³ + C
The antiderivative of 3x² is x³ because the derivative of x³ is 3x². The constant of integration, C, is added to represent the family of antiderivatives.
Consider the piecewise function f(x) = { x² for x < 1; 2x + 1 for x ≥ 1 }. Is f differentiable at x = 1?
f is not differentiable at x = 1
f is differentiable at x = 1 because the derivatives match
f is differentiable at x = 1 since both pieces are smooth
f is differentiable at x = 1 only if f(1) equals the average of the two limits
For a function to be differentiable at a point, it must be continuous there. In this case, f(x) is discontinuous at x = 1 because the left-hand limit (1) does not equal the function value from the right (3), so f is not differentiable at that point.
Find dy/dx using implicit differentiation for the equation x² + y² = 25.
dy/dx = -x/y
dy/dx = x/y
dy/dx = -y/x
dy/dx = y/x
Differentiating both sides of the equation x² + y² = 25 with respect to x gives 2x + 2y(dy/dx) = 0. Solving for dy/dx yields dy/dx = -x/y. The other options reflect common mistakes in implicit differentiation.
Evaluate the limit: lim (x → 0) (sin(5x))/x.
5
0
Undefined
1/5
Recognizing the standard limit lim (x → 0) (sin(kx))/x = k leads directly to the answer, where k is 5 in this case. The other options do not reflect the proper use of this limit property.
Let f(x) = ∫₀^(x²) sin(t) dt. Find f'(x).
2x sin(x²)
sin(x²)
2 sin(x)
x sin(x)
Using the Fundamental Theorem of Calculus in conjunction with the chain rule, differentiate the upper limit x² to obtain 2x and multiply by sin(x²). This results in f'(x) = 2x sin(x²).
Determine the derivative of the inverse function of f(x) = x³ + x.
(f❻¹)'(y) = 1 / (3(f❻¹(y))² + 1)
(f❻¹)'(y) = 3(f❻¹(y))² + 1
(f❻¹)'(y) = 1 / (3y² + 1)
(f❻¹)'(y) = f'(y)
The derivative of the inverse function is given by (f❻¹)'(y) = 1 / f'(x) where x = f❻¹(y). For f(x) = x³ + x, we have f'(x) = 3x² + 1, so the correct expression is 1 / (3(f❻¹(y))² + 1).
0
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Study Outcomes

  1. Understand fundamental differentiation concepts and their applications.
  2. Apply differentiation rules to compute derivatives of various functions.
  3. Analyze the behavior of functions using derivative tests.
  4. Evaluate definite integrals to determine areas under curves.
  5. Interpret the relationship between derivatives and integrals in solving calculus problems.

AP Calculus AB Unit 1 Practice Test Cheat Sheet

  1. Master the Concept of Limits - Limits show what value a function gets closer to as its input zeroes in on a point, even if the function never actually reaches it. They form the backbone of continuity and the launch pad into derivatives, so mastering them early is a calculus hack. Dive in here: AP Calculus AB Unit 1 Review | Limits and Continuity
  2. Evaluate Limits Numerically, Graphically & Analytically - Tables help you spot patterns, graphs let you visualize behavior, and algebraic techniques sharpen your analytical prowess. Juggling all three boosts your toolkit when one method alone hits a dead end. Level up your skills at Unit 1 Limits and Continuity
  3. Grasp Continuity and Its Tests - A function is continuous if you can draw it without lifting your pencil; checking limits from both sides and matching the function's defined value ensures smooth behavior. Spotting holes, jumps, and asymptotes makes you a graphing guru. Learn how it all ties together at Unit 1 Review - Limits & Continuity | Fiveable
  4. Harness the Intermediate Value Theorem - If a function is continuous on [a, b], it must take on every value between f(a) and f(b). It's like guaranteeing you'll taste every flavor at an ice cream buffet - no skips allowed! See why this theorem is root-perfect here: Unit 1 Review - Limits & Continuity | Fiveable
  5. Explore Infinite Limits & Limits at Infinity - Watch functions skyrocket to ±∞ near vertical asymptotes or settle into horizontal asymptotes out at the horizon. Recognizing these behaviors is key to graphing like a pro and predicting long-term trends. Get the lowdown at Unit 1 Review - Limits & Continuity | Fiveable
  6. Apply Algebraic Properties to Simplify Limits - Use the sum, difference, product, and quotient rules to break complex limits into bite-sized problems. These shortcuts turn messy expressions into straightforward calculations, saving time and headaches. Watch the magic unfold: AP Calculus AB - 1.5 Determining Limits Using Algebraic Properties
  7. Squeeze (Sandwich) Theorem - When a tricky function hangs out between two friendlier ones with the same limit, it must share their limit too. This squeeze play is especially handy for taming oscillating or trigonometric functions. Get squeezed into success here: AP Calculus AB Unit 1 Review | Limits and Continuity
  8. Spot Different Types of Discontinuities - Jump, removable, infinite - each discontinuity tells a story about function behavior. Understanding them helps you predict weird graph twists and turns like a calculus detective. Explore the tales at Unit 1 Review - Limits & Continuity | Fiveable
  9. Nail One‑Sided Limits - Sometimes you need to peek from the left or just the right before a cliff edge in a graph. One-sided limits help you analyze behavior near sharp turns and gaps with precision. Check out the view from both sides: AP Calculus AB Unit 1 Review | Limits and Continuity
  10. Fix Removable Discontinuities - When a function has a hole (like a lost donut), you can often plug the gap by redefining a single point. This makes the function continuous and donut… I mean function whole again. Get the fill-in at Unit 1 Review - Limits & Continuity | Fiveable
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