Antiderivatives, and Integrals
as Accumulated Area

  1. Write down a formula for an antiderivative of each of these functions. Recall that a proposed antiderivative can be verified by differentiating it, checking that the result is identical to the original function.

    \( a'(x) = x^7\!+\!\tfrac{1}{x^7}\!+\!\sqrt[7]{x}\!+\!\tfrac{x}{7}\)
    \( b'(x) = \tan(x)\cos(x)\)
    \( c'(x) = \csc^2(x)\)
    \( \alpha'(x) = 5\cos\bigl(5x\bigr)\)
    \( \beta'(x) = 2x\cos\bigl(x^2\bigr)\)
    \( \gamma'(x) = x^7\cos\bigl(x^8\bigr)\)
  2. If \(f'(x) = x^5-x,\) and \(f(1) = 2,\) what’s a formula for \(f(x)?\)
  3. If \(g''(x) = x^2\!-\!3,\) and \(g'(6) = 9,\) and \(g(0) = 7,\) what’s a formula for \(g(x)?\)
  4. If \(h''(x) = \sqrt{x},\) \(h(1) = \tfrac{1}{2},\) and \(h(0) = -16,\) what must a formula for \(h(x)\) be?

  5. What are the exact values of the following definite integrals? (Hint: think of them each as an area under a curve.)

    \(\displaystyle\int\limits_3^7 x \,\mathrm{d}x \)
    \(\displaystyle\int\limits_{-1}^{5} |x-1|-1 \,\mathrm{d}x \)
    \(\displaystyle\int\limits_{-2}^2 \sqrt{4-x^2} \,\mathrm{d}x \)
    \(\displaystyle\int\limits_{-100}^{100} \arctan(x) \,\mathrm{d}x \)

  6. For this graph of a function \(f\) estimate the values of these definite integrals.

    \(\displaystyle\int\limits_{3}^{4} f(x) \,\mathrm{d}x \)
    \(\displaystyle\int\limits_{1}^{4.5} f(x) \,\mathrm{d}x \)
    \(\displaystyle\int\limits_{-3}^{5} f(x) \,\mathrm{d}x \)

  7. Suppose that \(\int_0^2 f(x)\,\mathrm{d}x = 7\) and that \(f\) is an even function (\(f(x) = f(-x)\)). What must the value of these definite integrals be?

    \(\displaystyle \int\limits_{0}^{2} -5f(x) \,\mathrm{d}x \)
    \(\displaystyle \int\limits_{2}^{0} f(x) \,\mathrm{d}x \)
    \(\displaystyle \int\limits_{0}^{2} 3x+f(x) \,\mathrm{d}x \)
    \(\displaystyle \int\limits_{-2}^{2} f(x) \,\mathrm{d}x \)
  8. Consider the semicircle of radius \(r\) centered at the origin that sits above the \(x\)-axis; its equation is \(x^2+y^2=r^2\) for \(y \geq 0.\) What is the average height of a point on this semicircle? Hint: it’s not just \(\tfrac{1}{2}r.\)

  9. A cyclist is wearing a heart rate monitor while competing in a fifty-mile bike race. They check the monitor, which tracks their heart rate measured in beats-per-minute (bpm), occasionally throughout the race.

    Minutes since Start 0 10 33 58 70 101 120
    Heart Rate (bpm) 66 73 80 88 97 68 87

    First briskly sketch a scatter plot of this data with your independent variable measured in minutes since the start of the race. Then, estimating, how many times did their heart beat in those first \(10\) minutes of the race? What about the first \(33\) minutes of the race? What about the first \(70\) minutes of the race? What about for the whole race?

  10. We often denote the \(n^\text{th}\) derivative of a function \(f\) as \(f^{(n)}.\) For example, \(f^{(13)}(x)\) is the thirteenth derivative of \(f\) — this is better notation than \(f'''''''''''''(x).\) Given that \({f^{(5)}(x) = 12x+1}\) what’s a possible formula for \(f(x)?\) What’s a possible formula for \(f(x)\) if we also require that \(f^{(3)}(2)=21?\)
  11. Write some computer code to evaluate a sum. That is, presented with a sum \[\sum_{i=a}^{b}\;f(i)\] write a for-loop that will evaluate the sum. Then, using a large value of the upper index \(b,\) use your code to approximate the value of these convergent infinite sums; each resulting value should be a familiar number. \[ \frac{1}{2} \!+\!\frac{1}{4} \!+\!\frac{1}{8} \!+\!\frac{1}{16} \!+\!\frac{1}{32} \!+\!\frac{1}{64} \!+\!\dotsb \qquad\quad \frac{4}{1} \!-\!\frac{4}{3} \!+\!\frac{4}{5} \!-\!\frac{4}{7} \!+\!\frac{4}{9} \!-\!\frac{4}{11} \!+\!\dotsb \]