Antiderivatives with initial conditions practice problems
Find the antiderivative, then use the initial condition to pin down the constant. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Solve the initial-value problem \( \displaystyle f'(x) = 3 x^{2} - 3 \cos{\left(x \right)} \), \( \displaystyle f(0) = -3 \).
Solve the initial-value problem \( \displaystyle f'(x) = 5 x^{3} - \frac{3}{x^{2}} \), \( \displaystyle f(1) = 0 \).
Solve \( \displaystyle f''(x) = - 4 x^{2} - 3 \cos{\left(x \right)} \) with \( \displaystyle f'(1) = -6 \) and \( \displaystyle f(1) = -2 \).
Solve \( \displaystyle f''(x) = x^{3} - 4 \sin{\left(x \right)} \) with \( \displaystyle f'(1) = 6 \) and \( \displaystyle f(1) = 1 \).
Solve the initial-value problem \( \displaystyle f'(x) = \sqrt{x} + 6 \), \( \displaystyle f(1) = -2 \).
Solve the initial-value problem \( \displaystyle f'(x) = 2 x^{2} - 3 \sin{\left(x \right)} \), \( \displaystyle f(1) = 4 \).
Solve \( \displaystyle f''(x) = - 2 \sqrt{x} + 4 x^{2} \) with \( \displaystyle f'(1) = 4 \) and \( \displaystyle f(1) = 2 \).
Solve \( \displaystyle f''(x) = 4 \cos{\left(x \right)} + 4 \) with \( \displaystyle f'(0) = 3 \) and \( \displaystyle f(0) = -4 \).
Solve \( \displaystyle f''(x) = 2 - \frac{1}{x^{2}} \) with \( \displaystyle f'(1) = 1 \) and \( \displaystyle f(1) = -5 \).
Solve \( \displaystyle f''(x) = 2 - \sin{\left(x \right)} \) with \( \displaystyle f'(1) = -3 \) and \( \displaystyle f(1) = -4 \).