Practice and review
Mixed practice
Choose the mathematical idea before calculating. These nine problems mix the course topics. Try each one unaided, or reveal two progressively more specific hints. Your final review distinguishes independent solutions from work to revisit. Progress is saved in this browser.
1. A missing point
1 / 9f(x)=\frac{x^2-4}{x-2},\quad x\ne2 The expression is defined for every real input except 2. We want to extend the graph through that missing input without a break.
- Compare the values approaching the missing input from either side.
- Factor x²−4 as (x−2)(x+2). Cancel only when x≠2.
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.
2. A changing signal
2 / 9S(t)=(3t+1)^2 A sensor reports S(t)=(3t+1)². Find its instantaneous rate, rather than its reported value.
- There is a function inside another function.
- Let u=3t+1. The outer derivative is 2u and du/dt=3.
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.
3. A changing ratio
3 / 9q(x)=\frac{x^2}{x+1} The output is a quotient. Both its numerator and denominator change with x.
- Dividing the two derivatives will not give the derivative of the quotient.
- Use (u′v−uv′)/v² with u=x² and v=x+1.
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.
4. The best value
4 / 9C(x)=x^2-4x+7,\quad 0\le x\le3 A cost model is C(x)=x²−4x+7 on the closed interval [0,3]. Find the lowest cost over the whole interval.
- Check interior critical points and both endpoints.
- Solve 2x−4=0, then evaluate C there and at 0 and 3.
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.
5. A short trip
5 / 9v(t)=t-2,\quad 0\le t\le4 An object has velocity v(t)=t−2 meters per second from t=0 to t=4 seconds. It changes direction during the trip.
- Total distance cannot cancel when direction changes.
- Split at t=2 and add the magnitudes of the two triangular areas.
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.
6. A weighted total
6 / 9I=\int_0^1 xe^x\,dx Accumulate the product of x and eˣ over [0,1]. A direct power rule does not apply to this product.
- Choose one factor to differentiate and one to integrate.
- Take u=x and dv=eˣ dx. Then evaluate xeˣ−eˣ at both bounds.
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.
7. A moving endpoint
7 / 9A(x)=\int_0^{x^2}(t+1)\,dt The upper limit of this accumulated quantity changes with x. The graph shows the integrand, not A itself.
- Differentiating accumulated area recovers the integrand at the endpoint.
- The endpoint is x², so its derivative 2x must also appear.
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.
8. A hollow object
8 / 91\le y\le3,\quad 0\le x\le2 Rotate the region 1≤y≤3 and 0≤x≤2 around the x-axis. The picture shows the generating region.
- The central hole removes volume.
- Subtract squared radii, then multiply the cross-sectional area by the length.
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.
9. A balance point
9 / 9\rho(x)=x,\quad 0\le x\le6 A rod lies on [0,6] with linear density ρ(x)=x. The right end is denser than the left. The graph shows density.
- Use a density-weighted average, not the geometric midpoint.
- Divide ∫xρ(x) dx by ∫ρ(x) dx over [0,6].
Explore the graph, then answer using the values in the question. Reveal a hint if you need a starting point.