Calculus - Practice Set 1 | Math AI

Q1

Differentiate: (a) f(x)=5x³−3x²+7x−2,

(b) g(x)=4√x − 3/x,

(c) h(x)=e^(2x).

Answer

(a) 15x²−6x+7 (b) 2/√x + 3/x² (c) 2e^(2x)

Q2

Find the equation of the tangent and normal to y=x²−3x+1 at the point where x=2.

Answer

Tangent: y=x−3; Normal: y=−x+1

Q3

Find:

(a) ∫(6x²−4x+3) dx,

(b) ∫e^(3x) dx,

(c) ∫(1/x) dx.

Answer

(a) 2x³−2x²+3x+C (b) e^(3x)/3+C (c) ln|x|+C

Q4

Evaluate: ∫₁⁴ (3x − 2) dx.

Answer

16.5

Q5

f(x) = 2x³ − 9x² + 12x − 4

(a) Find f'(x)

(b) Find all stationary points

(c) Classify them using the second derivative.

Answer

(a) 6x²−18x+12 (b) (1,1) max and (2,0) min (c) x=1 max, x=2 min

Q6

Use the trapezoidal rule with n=4 strips to estimate ∫₀² √(1+x²) dx.

Answer

≈2.977 (exact ≈3.056)

Q7

An object moves with velocity v(t) = 3t² − 6t + 2 m/s

(a) Find the acceleration at t=1

(b) Find the displacement from t=0 to t=3

(c) Find when v=0.

Answer

(a) 0 m/s² (b) 6 m (c) t≈0.42 s and t≈1.58 s

Q8

f(x) = x·eˣ

(a) Use the product rule to find f'(x)

(b) Find f''(x)

(c) Find the x-coordinate of any inflection point

(d) Sketch the key features.

Answer

(a) eˣ(1+x) (b) eˣ(2+x) (c) x=−2 (d) Min at x=−1

Q9

Use integration by substitution to find:

(a) ∫ 2x(x²+3)⁴ dx,

(b) ∫ sin(3x+1) dx,

(c) ∫ xe^(x²) dx.

Answer

(a) (x²+3)⁵/5+C (b) −cos(3x+1)/3+C (c) eˣ²/2+C

Q10

Find the exact area enclosed between y=x² and y=6x−x². Show all working.

Answer

9 square units

Q11

f(x) = (2x+1)/(x−3). Find:

(a) f'(x) using the quotient rule

(b) the equation of the tangent at x=5

(c) whether f is increasing or decreasing for x>3.

Answer

(a) −7/(x−3)² (b) y=−7x/4+47/4 (c) Strictly decreasing

Q12

Volume of revolution: the region bounded by y=√x, x=0, x=4, and the x-axis is rotated about the x-axis.

(a) Set up the integral.

(b) Evaluate exactly.

(c) Find volume when same region rotated about y-axis.

Answer

(a) V=π∫₀⁴x dx (b) 8π (c) 128π/5

Q13

A particle: s(t)=t³−6t²+9t+2 for t≥0

(a) Find velocity and acceleration

(b) When is particle at rest?

(c) Find total distance (not displacement) from t=0 to t=4

(d) When is acceleration=0?

Answer

(a) v=3t²−12t+9, a=6t−12 (b) t=1,3 (c) 12 m (d) t=2

Q14

Separable ODE: dy/dx = y·cos(x)

(a) Separate and integrate

(b) Find the particular solution with y(0)=3

(c) Find y(π/2)

(d) Describe the long-term behaviour.

Answer

(a) y=Ae^(sinx) (b) y=3e^(sinx) (c) 3e≈8.15 (d) y oscillates between ≈1.10 and 8.15

Q15

Find the dimensions of a rectangle with perimeter 40cm that has the maximum possible area. Use calculus to verify it is a maximum.

Answer

Square with side 10 cm; maximum area = 100 cm²

Q16

Euler's method: dy/dx=x+y, y(0)=1. Use step h=0.25 to estimate y(1).

Answer

y(1) ≈ 2.883 (exact: 2e−1≈4.436; Euler underestimates as f is concave up)

Q17

Find:

(a) ∫ sin²(x) dx using the identity cos(2x)=1−2sin²(x)

(b) ∫₀^(π/2) sin²(x) dx exactly

(c) ∫ x·sin(x) dx using integration by parts.

Answer

(a) x/2−sin(2x)/4+C (b) π/4 (c) −xcos(x)+sin(x)+C

Q18

The diagram shows the area between f(x)=x³−4x²+4x (blue) and g(x)=x (red), with two shaded regions

$Calculus1 study illustration$

(a) Find all x-coordinates where f(x)=g(x)

(b) Determine which function is greater in each interval

(c) Find the total enclosed area

(d) Set up and evaluate ∫₀¹(f−g)dx and ∫₁³(g−f)dx separately, then sum the absolute values.

Answer

(a) x=0,1,3 (b) f>g on [0,1]; g>f on [1,3] (c) 37/12 (d) 5/12+8/3=37/12

Q19

The slope field shows dy/dx=y−x². A particular solution with y(0)=3 is drawn.

$Calculus2 study illustration$

(a) Verify the solution y=eˣ+x²+2x+2 satisfies the ODE and initial condition

(b) Find y(2) and the gradient of the curve at x=2

(c) Use Euler's method with h=0.5 to estimate y(2) starting from y(0)=3. Compare with exact

(d) Find all equilibrium solutions (where dy/dx=0 always).

Answer

(a) Verified (b) y(2)=e²+10≈17.4, slope≈13.4 (c) Euler gives ≈13.03, error≈4.36 (d) No equilibrium

Q20

Volume of revolution: region bounded by y=√x+0.5, x-axis, x=0, x=3. The diagram shows a disk at x=2

$Calculus3 study illustration$

(a) Set up the integral for volume about x-axis

(b) Evaluate exactly

(c) Set up the integral for volume about y-axis

(d) Find the surface area of revolution about the x-axis using SA=2π∫₀³y√(1+(dy/dx)²)dx.

Answer

(a) π∫₀³(√x+0.5)²dx (b) π(5.25+3√3)≈31.1 (c) ≈57.2 (d) SA≈47.3

Q21

v(t)=3t²−12t+9 m/s (blue), a(t)=6t−12 m/s² (red) shown

$Calculus4 study illustration$

(a) Find when the particle momentarily stops. State direction of motion on each interval

(b) Find total distance from t=0 to t=5

(c) Find the maximum speed in [0,5]

(d) Show the displacement from t=0 to t=5 equals ∫₀⁵v(t)dt and compute it

(e) Find the position function s(t) if s(0)=0, and find when s is at its initial position again.

Answer

(a) t=1,3; right,left,right (b) 28 m (c) 24 m/s at t=5 (d) Displacement=20m (e) t=3

Q22

Optimization (hard): An open-topped rectangular box is made from a 20cm×30cm sheet of metal by cutting squares of side x from each corner and folding up.

(a) Show the volume is V(x)=x(20−2x)(30−2x)

(b) Expand and find V'(x)

(c) Find the exact value(s) of x that maximize V

(d) Show the second derivative confirms it is a maximum

(e) Find the dimensions and maximum volume.

Answer

(a) Shown (b) V'=12x²−200x+600 (c) x=(25−5√7)/3≈3.93 (d) V''<0 (e) V≈1056 cm³

Q23

Differential equation application: Newton's Law of Cooling: dT/dt=−k(T−20), where T is temperature(°C), t is time(min), 20°C is room temp. A cup of coffee cools from 90°C to 70°C in 5 minutes.

(a) Solve the ODE to get T(t)=20+Ae^(−kt)

(b) Use initial condition to find A

(c) Use T(5)=70 to find k exactly

(d) Find T(20)

(e) Find how long until T=30°C

(f) If the room temp changes to 25°C after t=10, set up the new IVP.

Answer

(a) T=20+Ae^(−kt) (b) A=70 (c) k=ln(7/5)/5 (d) ≈38.2°C (e) ≈49.1 min (f) dT/dt=−k(T−25), T(10)=known

Q24

Further integration:

(a) Find ∫ x²·ln(x) dx using integration by parts

(b) Evaluate ∫₁² x²·ln(x) dx exactly

(c) Find ∫ cos²(x)·sin(x) dx using substitution

(d) Find ∫₀^(π/2) cos²(x)·sin(x) dx

(e) Show that ∫₀^(π/2) sinⁿ(x)dx = (n−1)/n · ∫₀^(π/2) sinⁿ⁻²(x)dx and use it for n=4.

Answer

(a) x³lnx/3−x³/9+C (b) 8ln2/3−7/9 (c) −cos³x/3+C (d) 1/3 (e) ∫₀^(π/2)sin⁴x dx=3π/16

Q25

Coupled differential equations / Euler's method hybrid: A predator-prey model: dx/dt = 0.4x − 0.02xy (prey), dy/dt = −0.3y + 0.01xy (predator), where x=prey (hundreds), y=predator (tens). Initial: x(0)=40, y(0)=9

(a) Find the equilibrium points (dx/dt=dy/dt=0)

(b) Interpret the equilibrium biologically

(c) Use Euler's method with h=0.5 to estimate x(1) and y(1)

(d) Find dx/dt and dy/dt at t=0 and interpret

(e) This system is called the Lotka-Volterra model. Describe qualitatively what happens to x and y over time.

Answer

(a) (0,0) and (30,20) (b) Stable coexistence (c) x(1)≈49.1, y(1)≈10.1 (d) Both growing (e) Cyclic oscillations

UNIT 5: CALCULUS