I85PREP
⌘ K
CLEP Calculus study materials
CLEP Calculus study resources

CLEP Calculus practice questions

One question at a time. Build your CLEP Calculus understanding.

First 50 of 200 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Limits

Evaluate the limit: lim(x→3) (2x² − 5x + 1).

  1. 4 Correct answer.
  2. 10
  3. 22
  4. 16

Explanation

A polynomial is continuous everywhere, so the limit equals the function value at x = 3. Substituting gives 2(9) − 15 + 1 = 18 − 15 + 1 = 4.

Question 2Limits

Evaluate lim(x→2) (x² − 4)/(x − 2).

  1. undefined
  2. 0
  3. 2
  4. 4 Correct answer.

Explanation

The expression is 0/0 at x = 2, so factor the numerator as (x − 2)(x + 2). Canceling the common factor leaves x + 2, whose limit as x→2 is 4.

Question 3Limits

What is lim(x→∞) (3x² + 2x)/(x² − 7)?

  1. ∞
  2. 0
  3. 7
  4. 3 Correct answer.

Explanation

When numerator and denominator have the same degree, the limit at infinity is the ratio of leading coefficients. Here that ratio is 3/1 = 3.

Question 4Limits

Evaluate lim(x→0) (sin 5x)/x.

  1. 5 Correct answer.
  2. 1/5
  3. 1
  4. 0

Explanation

Using the standard limit lim(u→0) sin(u)/u = 1 with u = 5x, rewrite the expression as 5 · (sin 5x)/(5x). The limit is therefore 5 · 1 = 5.

Question 5Limits

The limit lim(x→4) f(x) exists and equals 7. Which statement must be true?

  1. f is continuous at x = 4
  2. f is defined at x = 4
  3. f(x) can be made arbitrarily close to 7 by taking x close to 4 Correct answer.
  4. f(4) = 7

Explanation

The limit describes the behavior of f near x = 4, not at x = 4 itself. A limit of 7 means the function values approach 7 as x approaches 4, regardless of the function's value there.

Question 6Limits

Evaluate lim(x→−2⁺) 1/(x + 2).

  1. +∞ Correct answer.
  2. 0
  3. 1/2
  4. −∞

Explanation

As x approaches −2 from the right, x + 2 is small and positive, so the reciprocal grows without bound in the positive direction. Hence the one-sided limit is +∞.

Question 7Limits

Evaluate lim(x→0) (√(x + 9) − 3)/x.

  1. 1/3
  2. 0
  3. 3
  4. 1/6 Correct answer.

Explanation

Multiply numerator and denominator by the conjugate √(x + 9) + 3 to get x/(x(√(x + 9) + 3)). Canceling x leaves 1/(√(x + 9) + 3), which tends to 1/(3 + 3) = 1/6.

Question 8Limits

What is lim(x→∞) (4x³ − x)/(2x³ + 5x²)?

  1. 0
  2. 2 Correct answer.
  3. 1/2
  4. ∞

Explanation

The numerator and denominator both have degree 3, so the limit at infinity equals the ratio of leading coefficients, 4/2 = 2. Lower-degree terms become negligible.

Question 9Limits

Let f(x) = (x² − 1)/(x − 1) for x ≠ 1 and f(1) = 3. Which statement describes f at x = 1?

  1. f has an infinite discontinuity at x = 1
  2. f has a jump discontinuity at x = 1
  3. f is continuous at x = 1
  4. f has a removable discontinuity at x = 1 Correct answer.

Explanation

The limit as x→1 of (x² − 1)/(x − 1) equals 2, but f(1) = 3, so the limit exists yet differs from the function value. This is a removable discontinuity because redefining f(1) as 2 would make it continuous.

Question 10Limits

Evaluate lim(x→0) (1 − cos x)/x².

  1. 1
  2. 0
  3. 2
  4. 1/2 Correct answer.

Explanation

Multiply numerator and denominator by 1 + cos x to obtain (sin²x)/(x²(1 + cos x)). Since (sin x)/x → 1 and 1 + cos x → 2, the limit is 1/2.

Question 11Limits

If lim(x→c) f(x) = 5 and lim(x→c) g(x) = −2, what is lim(x→c) [f(x) · g(x) + f(x)]?

  1. −15 Correct answer.
  2. −10
  3. −5
  4. 3

Explanation

The limit of a product is the product of limits, giving 5 · (−2) = −10, and the limit of the sum adds lim f = 5. Thus −10 + 5 = −5. Wait-recompute: the expression is f·g + f = −10 + 5 = −5.

Question 12Limits

Evaluate lim(x→2) (x² − 4)/(x² − 3x + 2).

  1. 4 Correct answer.
  2. 1
  3. 2
  4. 0

Explanation

Both numerator and denominator vanish at x = 2, so factor: numerator (x − 2)(x + 2), denominator (x − 2)(x − 1). Canceling x − 2 leaves (x + 2)/(x − 1), which tends to 4/1 = 4.

Question 13Limits

A function f satisfies f(1) = 2 and lim(x→1) f(x) = 4. Which conclusion is valid?

  1. f is continuous at x = 1
  2. f is differentiable at x = 1
  3. f is undefined at x = 1
  4. f is discontinuous at x = 1 Correct answer.

Explanation

Continuity at x = 1 requires f(1) to equal the limit as x→1. Since 2 ≠ 4, the function fails the continuity condition and is discontinuous there.

Question 14Limits

Evaluate lim(x→∞) (√(x² + 3x) − x).

  1. 3
  2. 3/2 Correct answer.
  3. 0
  4. ∞

Explanation

Multiply by the conjugate over itself: (x² + 3x − x²)/(√(x² + 3x) + x) = 3x/(√(x² + 3x) + x). Dividing by x gives 3/(√(1 + 3/x) + 1), which tends to 3/2.

Question 15Limits

The Squeeze Theorem is applied to a function g with 2 − x² ≤ g(x) ≤ 2 + x² for all x near 0. What is lim(x→0) g(x)?

  1. The limit cannot be determined
  2. 4
  3. 2 Correct answer.
  4. 0

Explanation

Both bounding functions tend to 2 as x→0, since x² → 0. Because g is trapped between two functions with the same limit, the Squeeze Theorem gives lim g(x) = 2.

Question 16Limits

Evaluate lim(x→0⁻) e^(1/x).

  1. ∞
  2. e
  3. 0 Correct answer.
  4. 1

Explanation

As x approaches 0 from the left, 1/x → −∞, and e raised to a quantity tending to −∞ approaches 0. Hence the left-hand limit is 0.

Question 17Limits

Let f(x) = (ax + 1) for x < 2 and f(x) = x² − 3 for x ≥ 2. For which value of a is f continuous at x = 2?

  1. 2 Correct answer.
  2. −1
  3. 1
  4. 0

Explanation

Continuity at x = 2 requires the left-hand limit 2a + 1 to equal the right-hand value 2² − 3 = 1. Solving 2a + 1 = 1 gives a = 0. Wait-this yields a = 0. Rechecking: 2a + 1 = 1 gives a = 0.

Question 18Limits

Evaluate lim(x→0) (tan x)/x.

  1. ∞
  2. 0
  3. 1 Correct answer.
  4. 1/2

Explanation

Write tan x = (sin x)/(cos x), so (tan x)/x = [(sin x)/x] · [1/(cos x)]. As x→0, the first factor tends to 1 and the second tends to 1, giving a limit of 1.

Question 19Limits

Evaluate lim(x→1) (x³ − 1)/(x² − 1).

  1. 1
  2. 3/2 Correct answer.
  3. 0
  4. ∞

Explanation

Factor both: x³ − 1 = (x − 1)(x² + x + 1) and x² − 1 = (x − 1)(x + 1). Canceling x − 1 leaves (x² + x + 1)/(x + 1), which at x = 1 equals 3/2.

Question 20Limits

Suppose lim(x→c) f(x) = 0 and g is a function with |g(x)| ≤ 5 for all x near c. What is lim(x→c) f(x)g(x)?

  1. It depends on the value of g(c)
  2. The limit does not exist
  3. 5
  4. 0 Correct answer.

Explanation

Because |g(x)| ≤ 5, we have |f(x)g(x)| ≤ 5|f(x)|, and 5|f(x)| → 0. By the Squeeze Theorem, the product tends to 0 regardless of the behavior of g beyond being bounded.

Question 21Differential Calculus

What is the limit of (3x^2 - 5x + 2)/(x - 1) as x approaches 1?

  1. 1 Correct answer.
  2. does not exist
  3. 3
  4. 0

Explanation

Factor the numerator as (3x - 2)(x - 1), cancel the common factor (x - 1), leaving 3x - 2. Substituting x = 1 gives 3(1) - 2 = 1. The limit exists even though the expression is undefined at x = 1.

Question 22Differential Calculus

For f(x) = x^3, what is f'(x)?

  1. 3x^2 Correct answer.
  2. x^2
  3. x^3/3
  4. 3x

Explanation

The power rule states that the derivative of x^n is n·x^(n-1). Applying it with n = 3 gives 3x^2.

Question 23Differential Calculus

A function f is continuous at x = a. Which statement must be true?

  1. f'(a) exists
  2. the limit of f(x) as x approaches a equals f(a) Correct answer.
  3. f is differentiable at x = a
  4. f has a local maximum at x = a

Explanation

Continuity at a point requires three things: f(a) is defined, the limit exists, and the limit equals f(a). Only the third condition is listed among the choices.

Question 24Differential Calculus

What is the derivative of sin(x) with respect to x?

  1. -sin(x)
  2. -cos(x)
  3. tan(x)
  4. cos(x) Correct answer.

Explanation

The standard derivative formula for the sine function is d/dx[sin x] = cos x. This follows from the limit definition using the special limits for sine and cosine.

Question 25Differential Calculus

Evaluate the limit of (sin(4x))/x as x approaches 0.

  1. 1
  2. 0
  3. 1/4
  4. 4 Correct answer.

Explanation

Rewrite the expression as 4·(sin(4x))/(4x). Since (sin u)/u approaches 1 as u approaches 0, the limit is 4·1 = 4.

Question 26Differential Calculus

What is the derivative of e^(2x)?

  1. 2e^(2x) Correct answer.
  2. e^(2x)/2
  3. e^(2x)
  4. 2x·e^(2x)

Explanation

By the chain rule, d/dx[e^u] = e^u·u'. With u = 2x, u' = 2, so the derivative is 2e^(2x).

Question 27Differential Calculus

The graph of f has a horizontal tangent line at x = c and f''(c) > 0. What does this indicate about f at x = c?

  1. an inflection point
  2. a local minimum Correct answer.
  3. a local maximum
  4. a vertical asymptote

Explanation

A horizontal tangent means f'(c) = 0, a critical point. If the second derivative is positive there, the graph is concave up, so the critical point is a local minimum by the second derivative test.

Question 28Differential Calculus

Which expression gives the slope of the tangent line to y = f(x) at x = a?

  1. the limit of (f(a+h) - f(a))/h as h approaches 0 Correct answer.
  2. f(a)
  3. the limit of (f(a+h) - f(a))/h as h approaches infinity
  4. f(a+h) - f(a)

Explanation

The derivative at a point is defined as the limit of the difference quotient (f(a+h) - f(a))/h as h approaches 0. This limit, when it exists, is the tangent slope.

Question 29Differential Calculus

Find the derivative of f(x) = x^2·cos(x).

  1. 2x·cos(x) - x^2·sin(x) Correct answer.
  2. -2x·sin(x)
  3. x^2·cos(x) - 2x·sin(x)
  4. 2x·cos(x) + x^2·sin(x)

Explanation

The product rule gives f'(x) = (x^2)'·cos x + x^2·(cos x)' = 2x·cos x + x^2·(-sin x). Simplifying yields 2x·cos x - x^2·sin x.

Question 30Differential Calculus

What is the derivative of (3x + 1)/(x - 2)?

  1. -3/(x - 2)^2
  2. 3/(x - 2)^2
  3. -7/(x - 2)^2 Correct answer.
  4. 7/(x - 2)^2

Explanation

By the quotient rule, the derivative is [(3)(x - 2) - (3x + 1)(1)]/(x - 2)^2. The numerator simplifies to 3x - 6 - 3x - 1 = -7, giving -7/(x - 2)^2.

Question 31Differential Calculus

Find dy/dx for y = (2x + 5)^4.

  1. (2x + 5)^3
  2. 8(2x + 5)^4
  3. 4(2x + 5)^3
  4. 8(2x + 5)^3 Correct answer.

Explanation

The chain rule gives dy/dx = 4(2x + 5)^3·(2), since the inner derivative of 2x + 5 is 2. Multiplying the constants yields 8(2x + 5)^3.

Question 32Differential Calculus

The position of a particle is s(t) = t^3 - 6t^2 + 9t (in meters, t in seconds). At what time is the particle at rest?

  1. t = 1 and t = 3 Correct answer.
  2. t = 2 only
  3. t = 0 and t = 2
  4. t = 3 only

Explanation

Velocity is s'(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3). The particle is at rest when velocity is zero, which occurs at t = 1 and t = 3.

Question 33Differential Calculus

Find the derivative of ln(5x) for x > 0.

  1. 5/x
  2. 1/x Correct answer.
  3. ln(5)/x
  4. 1/(5x)

Explanation

Using the chain rule, d/dx[ln(5x)] = (1/(5x))·5 = 1/x. This matches the fact that ln(5x) = ln 5 + ln x, whose derivative is 1/x.

Question 34Differential Calculus

Use implicit differentiation to find dy/dx for x^2 + y^2 = 25.

  1. -x/y Correct answer.
  2. x/y
  3. -x/(2y)
  4. -2x/y

Explanation

Differentiating both sides with respect to x gives 2x + 2y·(dy/dx) = 0. Solving for dy/dx yields dy/dx = -2x/(2y) = -x/y.

Question 35Differential Calculus

The graph of f(x) = x^3 - 3x is concave down on which interval?

  1. (-1, 1)
  2. (0, infinity)
  3. (-infinity, 0) Correct answer.
  4. (-infinity, infinity)

Explanation

Compute f''(x) = 6x. The graph is concave down where f''(x) < 0, which means 6x < 0, or x < 0. Thus the interval is (-infinity, 0).

Question 36Differential Calculus

Evaluate the limit of (x^2 - 9)/(x^2 - 3x) as x approaches 3.

  1. 3
  2. 2 Correct answer.
  3. 1
  4. does not exist

Explanation

Factor both parts: (x - 3)(x + 3) over x(x - 3). Cancel (x - 3) to get (x + 3)/x, then substitute x = 3 to obtain 6/3 = 2.

Question 37Differential Calculus

What is the equation of the tangent line to y = x^2 at the point (2, 4)?

  1. y = 4x - 4 Correct answer.
  2. y = 2x
  3. y = 4x - 8
  4. y = 4x + 4

Explanation

The derivative y' = 2x equals 4 at x = 2, so the tangent slope is 4. Using point-slope form, y - 4 = 4(x - 2), which simplifies to y = 4x - 4.

Question 38Differential Calculus

If f(x) = x·e^x, find f'(x).

  1. (x - 1)e^x
  2. (x + 1)e^x Correct answer.
  3. x·e^x
  4. e^x

Explanation

Apply the product rule: f'(x) = (1)·e^x + x·e^x = e^x(1 + x). This factors as (x + 1)e^x.

Question 39Differential Calculus

A differentiable function f has f(1) = 3 and f(4) = 9. Which value is guaranteed by the Mean Value Theorem for some c in (1, 4)?

  1. f'(c) = 2 Correct answer.
  2. f'(c) = 6
  3. f(c) = 6
  4. f''(c) = 2

Explanation

The Mean Value Theorem guarantees some c in (1, 4) with f'(c) equal to the average rate of change (f(4) - f(1))/(4 - 1) = (9 - 3)/3 = 2.

Question 40Differential Calculus

Find the derivative of f(x) = arctan(x).

  1. 1/sqrt(1 - x^2)
  2. 1/(1 + x^2) Correct answer.
  3. 1/(1 - x^2)
  4. sec^2(x)

Explanation

The standard derivative of the inverse tangent is 1/(1 + x^2). It comes from implicit differentiation of tan y = x, giving sec^2(y)·y' = 1 and y' = 1/(1 + x^2).

Question 41Differential Calculus

Let f(x) = x^3 - 3x^2 + 4. On the interval [-1, 4], what is the absolute maximum value of f?

  1. 4
  2. 20 Correct answer.
  3. -4
  4. 0

Explanation

Critical points satisfy f'(x) = 3x^2 - 6x = 3x(x - 2) = 0, giving x = 0 and x = 2. Evaluating f at critical points and endpoints: f(-1) = 0, f(0) = 4, f(2) = 0, f(4) = 20. The largest is 20.

Question 42Differential Calculus

Using the limit definition of the derivative, find f'(x) for f(x) = sqrt(x), where x > 0.

  1. 1/(2·sqrt(x)) Correct answer.
  2. 2·sqrt(x)
  3. sqrt(x)/2
  4. 1/sqrt(x)

Explanation

The difference quotient is (sqrt(x + h) - sqrt(x))/h. Multiplying numerator and denominator by the conjugate gives 1/(sqrt(x + h) + sqrt(x)), whose limit as h approaches 0 is 1/(2·sqrt(x)).

Question 43Differential Calculus

The curve defined by x^2 + xy + y^2 = 7 passes through the point (1, 2). What is the slope of the tangent line at that point?

  1. -1
  2. 4/5
  3. -4/5 Correct answer.
  4. -5/4

Explanation

Differentiate implicitly: 2x + y + x·y' + 2y·y' = 0. Solving gives y' = -(2x + y)/(x + 2y). At (1, 2) this is -(2 + 2)/(1 + 4) = -4/5.

Question 44Differential Calculus

A ladder 10 feet long leans against a wall, and its base slides away from the wall at 2 ft/s. How fast is the top of the ladder sliding down when the base is 6 feet from the wall?

  1. 2 ft/s
  2. 3/2 ft/s Correct answer.
  3. 4/3 ft/s
  4. 5/2 ft/s

Explanation

Let x be the base distance and y the height, so x^2 + y^2 = 100. Differentiating gives 2x·x' + 2y·y' = 0. When x = 6, y = 8, so y' = -(6·2)/8 = -3/2, meaning the top descends at 3/2 ft/s.

Question 45Differential Calculus

Suppose f is twice differentiable with f'(3) = 0 and f''(3) = 0, and f''(x) changes sign at x = 3. What is true at x = 3?

  1. f has a local maximum
  2. f has an inflection point Correct answer.
  3. f has a local minimum
  4. f has a vertical tangent

Explanation

A sign change in f'' at a point where the function is defined indicates a change in concavity, which defines an inflection point. The second derivative test is inconclusive when f''(3) = 0.

Question 46Differential Calculus

What is the derivative of the constant function f(x) = 7?

  1. 7
  2. 1
  3. 0 Correct answer.
  4. 7x

Explanation

The derivative of any constant function is zero because the graph is a horizontal line with slope 0 everywhere.

Question 47Differential Calculus

If f(x) = sin(x), what is f'(x)?

  1. -cos(x)
  2. sin(x)
  3. cos(x) Correct answer.
  4. -sin(x)

Explanation

The derivative of sine is cosine, a standard derivative rule: d/dx[sin x] = cos x.

Question 48Differential Calculus

What is the derivative of f(x) = x^5?

  1. x^4
  2. 4x^5
  3. 5x^5
  4. 5x^4 Correct answer.

Explanation

The power rule states that d/dx[x^n] = n x^(n-1). Applying it with n = 5 gives 5x^4.

Question 49Differential Calculus

What is the derivative of f(x) = 3x^2 + 2x - 1?

  1. 6x + 2 Correct answer.
  2. 3x + 2
  3. 6x - 1
  4. 6x^2 + 2

Explanation

Differentiate term by term using the power rule: d/dx[3x^2] = 6x, d/dx[2x] = 2, and d/dx[-1] = 0, giving 6x + 2.

Question 50Differential Calculus

What is the derivative of f(x) = ln(x) for x > 0?

  1. ln(x)/x
  2. x
  3. 1/ln(x)
  4. 1/x Correct answer.

Explanation

The derivative of the natural logarithm is the reciprocal function: d/dx[ln x] = 1/x for x > 0.