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Calculus 1 final exam review

20 practice problems across the topics a Calculus 1 final usually covers, in course order, each with a worked solution checked line by line. Your own exam may differ, so check your syllabus.

Try each problem on paper before opening its solution. If one gives you trouble, the topic link above it has more like it. How we verify

1. Limits by factoring

Problem 1.140

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{3}{2}} \frac{2 x + \left(2 x - 1\right)^{2} - 7}{2 x - 3} \).
  1. \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{2 x + \left(2 x - 1\right)^{2} - 7}{2 x - 3}\right) \]
    limitStarting with the limit of the function at x = 3/2.✓ Proved
  2. \[ = \lim_{x \to \frac{3}{2}^+}\left(\frac{4 x^{2} - 2 x - 6}{2 x - 3}\right) \]
    algebra simplify factorExpand the squared term. Combine like terms in the numerator. Factor out a 2 from the numerator.✓ Proved
  3. \[ = \lim_{x \to \frac{3}{2}^+}\left(\frac{\left(x + 1\right) \left(4 x - 6\right)}{2 x - 3}\right) \]
    factorFactor the quadratic expression in the numerator.✓ Proved
  4. \[ = \lim_{x \to \frac{3}{2}^+}\left(2 x + 2\right) \]
    cancel simplifyCancel the common factor (2*x - 3). Distribute the 2.✓ Proved
  5. \[ = 5 \]
    substitution algebra limitSubstitute x = 3/2 into the expression. Multiply 2 by 3/2. Final evaluation of the limit.✓ Proved
Answer \( 5 \)
The graph of f(x) = (2x + (2x - 1)² - 7)/(2x - 3). Two points slide along the curve toward x = 3/2, one from each side; their heights close in on 5.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{2 x + \left(2 x - 1\right)^{2} - 7}{2 x - 3}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{3}{2}^+}\left(\frac{\frac{d}{d x} \left(2 x + \left(2 x - 1\right)^{2} - 7\right)}{\frac{d}{d x} \left(2 x - 3\right)}\right) \]
    lhopitalApply l'Hôpital's rule✓ Proved
  3. \[ = \lim_{x \to \frac{3}{2}^+}\left(4 x - 1\right) \]
    simplify simplify simplifyCompute derivatives Expand the numerator Reduce the fraction✓ Proved
  4. \[ = 5 \]
    limitEvaluate the linear function at x = 3/2✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

2. Trigonometric limits

Problem 1.155

medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{\sin^{2}{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \).
  1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin^{2}{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}}\right) \]
    limit algebraStart with the original limit. Apply the power rule for quotients.✓ Proved
  2. \[ = \lim_{u \to 0^+}\left(\frac{\sin^{2}{\left(u \right)}}{u^{2}}\right) \]
    substitution algebraSubstitute u = 2*x + 1, which implies u -> 0 as x -> -1/2. Distribute the square to the numerator and denominator.✓ Proved
  3. \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \sin^{2}{\left(u \right)}}{\frac{d}{d u} u^{2}}\right) \]
    lhopitalApply L'Hopital's rule because the limit is an indeterminate form 0/0.✓ Proved
  4. \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)} \cos{\left(u \right)}}{u}\right) \]
    simplify simplify algebraCompute the derivatives. Cancel the common factor of 2. Split the expression into a product.✓ Proved
  5. \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right) \lim_{u \to 0^+} \cos{\left(u \right)} \]
    limit-lawApply the product rule for limits.✓ Proved
  6. \[ = 1 \]
    limit simplify simplifyEvaluate the first limit component. Evaluate the second limit component. Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = sin(2x + 1)²/(2x + 1)². Two points slide along the curve toward x = -1/2, one from each side; their heights close in on 1.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin^{2}{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}}\right) \]
    limit rewriteRewrite the fraction as a square of sin over its argument.✓ Proved
  2. \[ = \left(\lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin{\left(2 x + 1 \right)}}{2 x + 1}\right)\right)^{2} \]
    limit-lawApply the limit law to the inner fraction.✓ Proved
  3. \[ = 1 \]
    limit simplifySince sin(u)/u -> 1 as u -> 0. The limit evaluates to 1.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

3. Limits at infinity

Problem 1.100

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{2 x}{\sqrt{\left(2 x - 1\right)^{2} + 4}} \).
  1. \[ \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{\left(2 x - 1\right)^{2} + 4}}\right) \]
    limitStarting with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{4 x^{2} - 4 x + 5}}\right) \]
    algebra simplifyExpand the squared term inside the square root. Combine the constant terms inside the square root.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{2}{\sqrt{\frac{4 x^{2} - 4 x + 5}{x^{2}}}}\right) \]
    rewrite limit-lawDivide the numerator and denominator by x (noting x > 0 as x approaches infinity). Apply the limit law for the quotient of two functions.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{2}{\sqrt{4 - \frac{4}{x} + \frac{5}{x^{2}}}}\right) \]
    simplifyDistribute the division by x**2 inside the square root.✓ Proved
  5. \[ = \lim_{x \to \infty} 1 \]
    limit-law simplifyEvaluate the limits of the individual terms as x approaches infinity. Simplify the expression inside the square root.✓ Proved
  6. \[ = 1 \]
    simplify simplifyEvaluate the square root and the final division. Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = 2x/sqrt((2x - 1)² + 4). A point runs right along the curve; its height settles toward 1.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{\left(2 x - 1\right)^{2} + 4}}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{4 x^{2} - 4 x + 5}}\right) \]
    rewriteExpand the square inside the sqrt.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{2}{\sqrt{4 - \frac{4}{x} + \frac{5}{x^{2}}}}\right) \]
    factor cancelFactor x**2 out of the sqrt. Cancel the common factor x.✓ Proved
  4. \[ = 1 \]
    limit simplifyAs x→∞, the terms with 1/x vanish. Compute the numeric value.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

4. Continuity and types of discontinuity

Problem 1.189

hard
Find every point where \( \displaystyle f(x) = \frac{x^{2} - 25}{x^{2} - 7 x + 10} \) is discontinuous, and classify each discontinuity as removable, a jump, or infinite.
  1. A rational function is continuous wherever its denominator is not zero, so only the zeros of the denominator can be discontinuities.
    Reviewed
  2. \[ x^{2} - 7 x + 10 = \left(x - 5\right) \left(x - 2\right) \]
    Factor the denominator.✓ Proved
  3. \[ \frac{x^{2} - 25}{x^{2} - 7 x + 10} = \frac{x + 5}{x - 2} \]
    The factor (x − 5) cancels.✓ Proved
  4. \[ \lim_{x \to 5^+}\left(\frac{x^{2} - 25}{x^{2} - 7 x + 10}\right) = \frac{10}{3} \]
    At x = 5 the limit exists, but f(5) is undefined: a removable discontinuity.✓ Proved
  5. \[ \left. x + 5 \right|_{\substack{ x=2 }} = 7 \]
    At x = 2 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved
Answer \( \text{removable at } x = 5;\ \text{infinite at } x = 2 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

5. The derivative from the limit definition

Problem 2.1526

medium Video
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = \frac{3}{x + 2} \), then find \( \displaystyle f'(1) \).
  1. By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
    Reviewed
  2. \[ \frac{3}{h + x + 2} \]
    Write out f(x + h).Reviewed
  3. \[ \frac{\frac{3}{h + x + 2} - \frac{3}{x + 2}}{h} = - \frac{3}{\left(x + 2\right) \left(h + x + 2\right)} \]
    Combine and cancel the factor h.✓ Proved
  4. \[ \lim_{h \to 0^+}\left(- \frac{3}{\left(x + 2\right) \left(h + x + 2\right)}\right) = - \frac{3}{\left(x + 2\right)^{2}} \]
    Now h → 0 by direct substitution.✓ Proved
  5. \[ \left. - \frac{3}{\left(x + 2\right)^{2}} \right|_{\substack{ x=1 }} = - \frac{1}{3} \]
    At x = 1.✓ Proved
Answer \( f'(x) = - \frac{3}{\left(x + 2\right)^{2}},\ f'(1) = - \frac{1}{3} \)
Top: the graph of f = 3/(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3/(x + 2)².

✓ Nihil obstat Lines: 3 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

6. Product rule

Problem 2.1839

hard Beautiful Video
Differentiate \( \displaystyle f(x) = 5 x e^{x} - 5 e^{x} \).
  1. \[ \frac{d}{d x} \left(5 x e^{x} - 5 e^{x}\right) \]
    sumStart with the derivative of the entire expression.✓ Proved
  2. \[ = \frac{d}{d x} 5 x e^{x} - \frac{d}{d x} 5 e^{x} \]
    sumApply the sum rule to differentiate each term separately.✓ Proved
  3. \[ = 5 \frac{d}{d x} x e^{x} - 5 \frac{d}{d x} e^{x} \]
    constant-multiple algebraFactor out the constant 5 from both terms. Factor out the constant 5.✓ Proved
  4. \[ = 5 x \frac{d}{d x} e^{x} + 5 e^{x} \frac{d}{d x} x - 5 \frac{d}{d x} e^{x} \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = 5 x \frac{d}{d x} e^{x} + 5 e^{x} - 5 \frac{d}{d x} e^{x} \]
    derivative algebraDifferentiate x with respect to x. Distribute the derivative operator.✓ Proved
  6. \[ = 5 \left(x - 1\right) \frac{d}{d x} e^{x} + 5 e^{x} \]
    algebraFactor out the common derivative term.✓ Proved
  7. \[ = 5 \left(x - 1\right) e^{x} + 5 e^{x} \]
    exponentialDifferentiate the exponential function exp(x).✓ Proved
  8. \[ = 5 x e^{x} \]
    algebra simplifyDistribute the term (x - 1). Combine like terms to get the final result.✓ Proved
Answer \( 5 x e^{x} \)
Top: the graph of f = 5x·e^(x) - 5e^(x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x·e^(x).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(5 x e^{x} - 5 e^{x}\right) \]
    derivative✓ Proved
  2. \[ = 5 \frac{d}{d x} x e^{x} - 5 \frac{d}{d x} e^{x} \]
    product✓ Proved
  3. \[ = 5 x e^{x} \]
    chain sum simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

7. Quotient rule

Problem 2.441

hard Video
Differentiate \( \displaystyle f(x) = \frac{\sqrt{4 x + 1}}{4 x + 2} \).
  1. \[ \frac{d}{d x} \frac{\sqrt{4 x + 1}}{4 x + 2} \]
    rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
  2. \[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \sqrt{4 x + 1}}{4 x + 2} \]
    product rewriteApply the product rule. Rewrite the square root as a fractional power.✓ Proved
  3. \[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
    chainApply the chain rule to the first term.✓ Proved
  4. \[ = - \frac{4 \sqrt{4 x + 1}}{\left(4 x + 2\right)^{2}} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
    chain constant-multiple algebraApply the chain rule to the second term. Simplify the constants. Distribute the exponents.✓ Proved
  5. \[ = - \frac{8 x}{\sqrt{4 x + 1} \left(4 x + 2\right)^{2}} \]
    algebra algebra algebra simplifyFind a common denominator. Simplify the numerator. Distribute the 4. Combine like terms.✓ Proved
Answer \( - \frac{2 x}{\left(2 x + 1\right)^{2} \sqrt{4 x + 1}} \)
Top: the graph of f = sqrt(4x + 1)/(4x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -4sqrt(4x + 1)/(4x + 2)² + 2/(sqrt(4x + 1)·(4x + 2)).
A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \frac{\sqrt{4 x + 1}}{4 x + 2} \]
    derivative rewrite rewriteStart by identifying the function to differentiate. Rewrite the square root as a power of 1/2 for easier handling. Rewrite the division as multiplication by a negative power.✓ Proved
  2. \[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \sqrt{4 x + 1}}{4 x + 2} \]
    productApply the product rule to the two factors.✓ Proved
  3. \[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \left(4 x + 1\right)}{2 \sqrt{4 x + 1} \left(4 x + 2\right)} \]
    chainApply the chain rule to the first factor (4*x + 1)**(1/2).✓ Proved
  4. \[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
    chainApply the chain rule to the second factor (4*x + 2)**(-1).✓ Proved
  5. \[ = - \frac{\sqrt{4 x + 1} \frac{d}{d x} \left(4 x + 2\right)}{\left(4 x + 2\right)^{2}} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
    chainApply the chain rule to differentiate the inner function 4*x + 2.✓ Proved
  6. \[ = - \frac{4 \sqrt{4 x + 1}}{\left(4 x + 2\right)^{2}} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
    chain simplify rewrite rewriteDifferentiate the inner linear function 4*x + 2 to get 4. Simplify the constants: (1/2)*4 = 2 and -1*4 = -4. Rewrite negative exponents as fractions. Rewrite the fractional exponent as a square root.✓ Proved
  7. \[ = - \frac{8 x}{\sqrt{4 x + 1} \left(4 x + 2\right)^{2}} \]
    algebra algebra simplify simplifyFind a common denominator to combine the two terms. Expand the numerator terms. Combine like terms in the numerator: 8x - 16x = -8x and 4 - 4 = 0. Final simplified form of the derivative.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

8. Chain rule

Problem 2.187

hard Beautiful Video
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2} \).
  1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2} \]
    constant-multiplePull out the constant factor.✓ Proved
  3. \[ = - \frac{\sqrt{2} \left(e^{x} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x}\right)}{2} \]
    productApply the product rule to the remaining terms.✓ Proved
  4. \[ = - \frac{\sqrt{2} \left(e^{x} \cos{\left(x + \frac{\pi}{4} \right)} + e^{x} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} \right)}\right)}{2} \]
    derivativeDifferentiate exp(x).✓ Proved
  5. \[ = - \frac{\sqrt{2} \left(- e^{x} \sin{\left(x + \frac{\pi}{4} \right)} + e^{x} \cos{\left(x + \frac{\pi}{4} \right)}\right)}{2} \]
    trig algebraDifferentiate cos(x + pi/4). Simplify the expression inside the parentheses.✓ Proved
  6. \[ = - \frac{\sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{4} \right)}\right) e^{x}}{2} \]
    algebraFactor out exp(x).✓ Proved
  7. \[ = e^{x} \sin{\left(x \right)} \]
    rewrite algebra algebra simplify algebra algebra simplifyExpand the trigonometric terms using angle addition formulas. Substitute the value of cos(pi/4) and sin(pi/4). Distribute the values. Combine like terms inside the parentheses. Simplify the coefficient of sin(x). Multiply the negative signs. Simplify the final constant.✓ Proved
Answer \( e^{x} \sin{\left(x \right)} \)
Top: the graph of f = -sqrt(2)·e^(x)·cos(x + pi/4)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sqrt(2)·e^(x)·sin(x + pi/4)/2 - sqrt(2)·e^(x)·cos(x + pi/4)/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}\right) \]
    derivative rewritePull out the constant factor -sqrt(2)/2✓ Proved
  2. \[ = - \frac{\sqrt{2} \left(e^{x} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x}\right)}{2} \]
    productApply the product rule to exp(x)*cos(x + pi/4)✓ Proved
  3. \[ = - \frac{\sqrt{2} \left(e^{x} \cos{\left(x + \frac{\pi}{4} \right)} + e^{x} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} \right)}\right)}{2} \]
    derivativeDifferentiate exp(x) to get exp(x)✓ Proved
  4. \[ = - \frac{\sqrt{2} \left(- e^{x} \sin{\left(x + \frac{\pi}{4} \right)} + e^{x} \cos{\left(x + \frac{\pi}{4} \right)}\right)}{2} \]
    derivativeDifferentiate cos(x + pi/4) to get -sin(x + pi/4)✓ Proved
  5. \[ = - \frac{\sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{4} \right)}\right) e^{x}}{2} \]
    algebra simplifyFactor out exp(x) and combine terms Final simplified derivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

9. Implicit differentiation

Problem 2.755

medium
The curve \( \displaystyle x^{3} + 2 x y + y^{3} = -2 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
  1. \[ -2 \]
    The point is on the curve.Reviewed
  2. Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.
    Reviewed
  3. \[ \frac{d}{d x} \left(x^{3} + 2 x Y{\left(x \right)} + Y^{3}{\left(x \right)}\right) = 3 x^{2} + \left(2 x + 3 Y^{2}{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} + 2 Y{\left(x \right)} \]
    Every y term picks up a factor dy/dx.✓ Proved
  4. \[ \frac{- 3 x^{2} - 2 y}{2 x + 3 y^{2}} \]
    Solve for dy/dx: minus F_x over F_y.Reviewed
  5. \[ - \frac{1}{5} \]
    At the point.Reviewed
Answer \( \frac{dy}{dx} = \frac{- 3 x^{2} - 2 y}{2 x + 3 y^{2}}, \quad \left.\frac{dy}{dx}\right|_{(1,-1)} = - \frac{1}{5} \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

10. Tangent lines

Problem 2.725

medium Video
Find an equation of the tangent line to \( \displaystyle y = 2 x + \sin{\left(x \right)} \) at \( \displaystyle x = \pi \).
  1. The tangent line passes through the point (a, f(a)) and has slope f'(a).
    Reviewed
  2. \[ \left. 2 x + \sin{\left(x \right)} \right|_{\substack{ x=\pi }} = 2 \pi \]
    The point of tangency.✓ Proved
  3. \[ \frac{d}{d x} \left(2 x + \sin{\left(x \right)}\right) = \cos{\left(x \right)} + 2 \]
    Differentiate.✓ Proved
  4. \[ \left. \cos{\left(x \right)} + 2 \right|_{\substack{ x=\pi }} = 1 \]
    The slope at the point.✓ Proved
  5. \[ x + \pi \]
    Point-slope form, then simplify.Reviewed
Answer \( y = x + \pi \)
The graph of y = 2x + sin(x) near x = pi. A secant line through the point (pi, 2pi) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at 1, giving y = x + pi.

✓ Nihil obstat Lines: 3 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

11. Related rates

Problem 3.92

medium
Car A drives east from an intersection at 59 mph and car B drives north from it at 57 mph. How fast is the distance between them changing when A is 9 mi and B is 12 mi from the intersection?
  1. With A at distance a and B at distance b, the distance between them satisfies D² = a² + b².
    Reviewed
  2. Differentiate: 2D dD/dt = 2a da/dt + 2b db/dt.
    Reviewed
  3. \[ \left. \sqrt{x^{2} + y^{2}} \right|_{\substack{ x=9\\ y=12 }} = 15 \]
    D at that moment.✓ Proved
  4. \[ \left. \frac{59 x + 57 y}{\sqrt{x^{2} + y^{2}}} \right|_{\substack{ x=9\\ y=12 }} = 81 \]
    dD/dt = (a·da/dt + b·db/dt)/D with a = 9, b = 12.✓ Proved
Answer \( 81 \ \text{mph} \approx 81\ \text{mph} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

12. Linear approximation

Problem 3.17

easy
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 49 \) to estimate \( \displaystyle f(\frac{249}{5}) \).
  1. The linearization at a is L(x) = f(a) + f'(a)(x − a).
    Reviewed
  2. \[ \left. \sqrt{x} \right|_{\substack{ x=49 }} = 7 \]
    f(a).✓ Proved
  3. \[ \left. \frac{d}{d x} \sqrt{x} \right|_{\substack{ x=49 }} = \frac{1}{14} \]
    f'(a).✓ Proved
  4. \[ \frac{x}{14} + \frac{7}{2} \]
    The linearization.Reviewed
  5. \[ \frac{247}{35} \]
    Evaluate at x = \frac{249}{5}.Reviewed
Answer \( L(\frac{249}{5}) = \frac{247}{35} \approx 7.05714 \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

13. Absolute extrema on a closed interval

Problem 3.46

hard Video
Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} + \frac{9 x^{2}}{2} + 6 x - 1 \) on \( \displaystyle [-4, 1] \).
  1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
    Reviewed
  2. \[ \frac{d}{d x} \left(x^{3} + \frac{9 x^{2}}{2} + 6 x - 1\right) = \left(x + 2\right) \left(3 x + 3\right) \]
    Differentiate and factor.✓ Proved
  3. Critical numbers inside [-4, 1]: -2, -1.
    Reviewed
  4. \[ \left. x^{3} + \frac{9 x^{2}}{2} + 6 x - 1 \right|_{\substack{ x=-4 }} = -17 \]
    f(-4).✓ Proved
  5. \[ \left. x^{3} + \frac{9 x^{2}}{2} + 6 x - 1 \right|_{\substack{ x=-2 }} = -3 \]
    f(-2).✓ Proved
  6. \[ \left. x^{3} + \frac{9 x^{2}}{2} + 6 x - 1 \right|_{\substack{ x=-1 }} = - \frac{7}{2} \]
    f(-1).✓ Proved
  7. \[ \left. x^{3} + \frac{9 x^{2}}{2} + 6 x - 1 \right|_{\substack{ x=1 }} = \frac{21}{2} \]
    f(1).✓ Proved
  8. The largest value is 21/2 and the smallest is -17.
    Reviewed
Answer \( \text{max } \frac{21}{2} \text{ at } x=1;\ \text{min } -17 \text{ at } x=-4 \)
The graph of f(x) = x³ + 9x²/2 + 6x - 1 on [-4, 1]. The candidates are the endpoints and the critical points inside; the highest is 21/2 and the lowest -17.

✓ Nihil obstat Lines: 5 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

14. Increasing, decreasing and concavity

Problem 3.61

medium Video
For \( \displaystyle f(x) = 3 x^{3} + 18 x^{2} + 27 x \), find the intervals where f is increasing or decreasing, where it is concave up or down, and its inflection points.
  1. \[ \frac{d}{d x} \left(3 x^{3} + 18 x^{2} + 27 x\right) = \left(x + 3\right) \left(9 x + 9\right) \]
    f' factored.✓ Proved
  2. f' > 0 outside [-3, -1] and f' < 0 between them (a positive quadratic with roots -3, -1).
    Reviewed
  3. \[ \frac{d^{2}}{d x^{2}} \left(3 x^{3} + 18 x^{2} + 27 x\right) = 18 x + 36 \]
    f''.✓ Proved
  4. \[ \left. 18 \left(x + 2\right) \right|_{\substack{ x=-2 }} = 0 \]
    f'' = 0 at x = -2, where it changes sign.✓ Proved
  5. f'' < 0 to the left of -2 (concave down) and f'' > 0 to the right (concave up).
    Reviewed
Answer \( \uparrow (-\infty,-3) \cup (-1,\infty);\ \downarrow (-3,-1);\ \text{concave down } (-\infty,-2),\ \text{up } (-2,\infty);\ \text{inflection at } x=-2 \)
Top: the graph of f = 3x³ + 18x² + 27x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 9x² + 36x + 27. The tangent is flat at x = -3, -1.

✓ Nihil obstat Lines: 3 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

15. Optimization

Problem 3.77

medium
A rectangular field along a straight river needs no fence on the river side. With 120 m of fencing, what is the largest area that can be enclosed?
  1. Let x be the two sides perpendicular to the river; the side parallel is 120 − 2x. Area A(x) = x(120 − 2x), 0 < x < 60.
    Reviewed
  2. \[ \frac{d}{d x} x \left(120 - 2 x\right) = 120 - 4 x \]
    A'(x).✓ Proved
  3. \[ \left. 120 - 4 x \right|_{\substack{ x=30 }} = 0 \]
    A'(x) = 0 at x = 30.✓ Proved
  4. \[ \frac{d^{2}}{d x^{2}} x \left(120 - 2 x\right) = -4 \]
    A'' < 0: a maximum.✓ Proved
  5. \[ \left. x \left(120 - 2 x\right) \right|_{\substack{ x=30 }} = 1800 \]
    30 m by 60 m.✓ Proved
Answer \( 1800 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

16. L'Hôpital's rule

Problem 1.525

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{- 3 x + e^{3 x - 1}}{\left(3 x - 1\right)^{2}} \).
  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{- 3 x + e^{3 x - 1}}{\left(3 x - 1\right)^{2}}\right) \]
    limit algebraDefine the limit from the right. Rearrange the numerator.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(- 3 x + e^{3 x - 1}\right)}{\frac{d}{d x} \left(3 x - 1\right)^{2}}\right) \]
    lhopitalApply L'Hopital's rule since the limit is 0/0.✓ Proved
  3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{3 e^{3 x - 1} - 3}{18 x - 6}\right) \]
    simplify algebra factorCompute the derivatives. Simplify the denominator. Factor out 3 from the numerator.✓ Proved
  4. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{e^{3 x - 1} - 1}{6 x - 2}\right) \]
    algebraSimplify the fraction.✓ Proved
  5. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(e^{3 x - 1} - 1\right)}{\frac{d}{d x} \left(6 x - 2\right)}\right) \]
    lhopitalApply L'Hopital's rule again.✓ Proved
  6. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{e^{3 x - 1}}{2}\right) \]
    simplify algebra algebraCompute the derivatives. Simplify the fraction. Reduce the fraction.✓ Proved
  7. \[ = \frac{1}{2} \]
    limit algebra simplifyEvaluate the limit by substitution. Simplify the exponent. Final evaluation.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-3x + e^(3x - 1))/(3x - 1)². Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 1/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{- 3 x + e^{3 x - 1}}{\left(3 x - 1\right)^{2}}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{3 e^{3 x - 1} - 3}{18 x - 6}\right) \]
    lhopitalApply l'Hôpital's rule once✓ Proved
  3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{e^{3 x - 1}}{2}\right) \]
    lhopital simplifyApply l'Hôpital's rule again Simplify the constant factor✓ Proved
  4. \[ = \frac{1}{2} \]
    limitEvaluate the remaining limit✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

17. Basic antiderivatives

Problem 4.1492

medium Video
Find \( \displaystyle \int \sqrt{3 x + 1} \, dx \). (Omit the constant of integration.)
  1. \[ \int \sqrt{3 x + 1}\, dx \]
    integral algebraStart with the integral of the function. Rewrite the square root as a power.✓ Proved
  2. \[ = \int \sqrt{3} \sqrt{x + \frac{1}{3}}\, dx \]
    algebraFactor out the constant 3 from the linear term.✓ Proved
  3. \[ = \sqrt{3} \int \sqrt{x + \frac{1}{3}}\, dx \]
    linearityMove the constant outside the integral.✓ Proved
  4. \[ = \frac{2 \sqrt{3} \left(x + \frac{1}{3}\right)^{\frac{3}{2}}}{3} \]
    antiderivative algebra simplify algebraApply the power rule for integration. Group the coefficients. Simplify 3**(1/2)/3 to 1/3**(1/2). Rewrite the inner term to match the original form.✓ Proved
  5. \[ = \frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9} \]
    algebra algebra simplify simplifyDistribute the power to the numerator and denominator. Combine the denominator terms. Simplify 3**(1/2) * 3**(3/2) to 3**2. Final simplification of the coefficient.✓ Proved
Answer \( \frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9} + C \)
Top: the graph of F = 2(3x + 1)^(3/2)/9. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sqrt(3x + 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \sqrt{3 x + 1}\, dx \]
    integral✓ Proved
  2. \[ = \frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9} \]
    antiderivativeApply the power rule for integrals.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

18. u-substitution

Problem 4.735

medium Video
Find \( \displaystyle \int \left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)} \, dx \). (Omit the constant of integration.)
  1. \[ \int \left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
    integralStart with the integral of the given function.✓ Proved
  2. \[ = \int \left(x + 1\right) \cos{\left(x^{2} + 2 x + 1 \right)}\, dx \]
    algebraExpand the squared term inside the cosine.✓ Proved
  3. \[ = \int \left(x \cos{\left(x^{2} + 2 x + 1 \right)} + \cos{\left(x^{2} + 2 x + 1 \right)}\right)\, dx \]
    linearityDistribute the (x + 1) term.✓ Proved
  4. \[ = \int \left(x \cos{\left(\left(x + 1\right)^{2} \right)} + \cos{\left(\left(x + 1\right)^{2} \right)}\right)\, dx \]
    algebraRewrite the expanded polynomial back in its squared form for clarity.✓ Proved
  5. \[ = \int x \cos{\left(\left(x + 1\right)^{2} \right)}\, dx + \int \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
    linearity algebraSplit the integral into two parts. Rewrite x as (x + 1) - 1 to facilitate substitution.✓ Proved
  6. \[ = \int \left(\left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)} - \cos{\left(\left(x + 1\right)^{2} \right)}\right)\, dx + \int \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
    algebraDistribute the cosine term.✓ Proved
  7. \[ = \int \left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
    simplifyThe extra terms cancel out.✓ Proved
  8. \[ = \int \frac{\left(2 x + 2\right) \cos{\left(\left(x + 1\right)^{2} \right)}}{2}\, dx \]
    rewriteExpress the integrand as a product involving the derivative of the inner function.✓ Proved
  9. \[ = \int \left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
    simplifySimplify the coefficient.✓ Proved
  10. \[ = \frac{\sin{\left(\left(x + 1\right)^{2} \right)}}{2} \]
    antiderivativeApply the substitution rule for the integral.✓ Proved
Answer \( \frac{\sin{\left(\left(x + 1\right)^{2} \right)}}{2} + C \)
Top: the graph of F = sin((x + 1)²)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x + 1)·cos((x + 1)²).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \int \left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
    integral rewrite✓ Proved
  2. \[ = \frac{\int \left(2 x + 2\right) \cos{\left(\left(x + 1\right)^{2} \right)}\, dx}{2} \]
    linearity✓ Proved
  3. \[ = \frac{\sin{\left(\left(x + 1\right)^{2} \right)}}{2} \]
    antiderivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

19. Definite integrals

Problem 4.1

medium Video
Evaluate \( \displaystyle \int_{-1}^{2} x + e^{2 x} \, dx \).
  1. By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
    Reviewed
  2. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} + \frac{e^{2 x}}{2}\right) = x + e^{2 x} \]
    F(x) = x**2/2 + exp(2*x)/2 is an antiderivative: its derivative is the integrand.✓ Proved
  3. \[ - \left. \frac{x^{2}}{2} + \frac{e^{2 x}}{2} \right|_{\substack{ x=-1 }} + \left. \frac{x^{2}}{2} + \frac{e^{2 x}}{2} \right|_{\substack{ x=2 }} = \frac{-1 + \left(3 + e^{4}\right) e^{2}}{2 e^{2}} \]
    F(b) − F(a).✓ Proved
Answer \( \frac{-1 + \left(3 + e^{4}\right) e^{2}}{2 e^{2}} \)
The graph of f(x) = x + e^(2x) from x = -1 to x = 2. Rectangles under the curve get thinner until they fill the area, -e^(-2)/2 + 3/2 + e^(4)/2. Area below the x-axis counts as negative.

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

20. Fundamental Theorem of Calculus, Part 1

Problem 4.17

medium
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{\sqrt{x}} e^{- t^{2}} \, dt \).
  1. By FTC Part 1 and the chain rule, d/dx ∫ from a to u(x) of g(t) dt = g(u(x)) u'(x).
    Reviewed
  2. \[ \frac{d}{d x} \sqrt{x} = \frac{1}{2 \sqrt{x}} \]
    u'(x).✓ Proved
  3. \[ \frac{e^{- x}}{2 \sqrt{x}} \]
    g(u(x)) u'(x).Reviewed
Answer \( \frac{e^{- x}}{2 \sqrt{x}} \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

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